Firstly, it's worth mentioning that the wording "or a similar spell of equal or higher level" and "or another spell of equal or greater level" is likely future-proofing to a certain degree, in case they bring out something like (or someone homebrews something like) "greater dispel magic" or something in the future, then if this specifically said "This layer is destroyed by dispel magic." then RAW, greater dispel magic wouldn't work, even though it clearly should.
Nonetheless, some of these layers have existing spells that would bring down the layers in question besides using the examples given. Let's tackle them one by one:
Orange Layer
This layer "is destroyed by a strong wind." A strong wind? Well, firstly, I'll include Joel Harmon's comment:
Incidentally, "Strong Wind" is a term defined on DMG 110. It describes some effects on flames, fog, and fliers, but doesn't have any hard numbers.
Unfortunately, as the comment says, there are no "hard number"; this appears to be describing an environmental effect, so yes, if someone cast prismatic wall outside on a very windy day, a DM could rule that this layer might go down immediately, but in the middle of the dungeon, that's not going to happen. So we need to use magic.
The gust cantrip isn't going to cut it; gust of wind, wind wall, warding wind and whirlwind, on the other hand, all create strong enough wind to take this wall down.
From gust of wind (PHB, pg. 248):
A line of strong wind 60 feet long at 10 feet wide blasts from you in a direction you choose for the spell's duration.
From wind wall (PHB, pg. 288):
A wall of strong wind rises from the ground at a point you choose within range.
From warding wind (XGtE, pg. 170):
A strong wind (20 miles per hour) blow around you in a 10-foot radius and moves with you, remaining centred on you.
From whirlwind (XGtE, pg. 171):
A whirlwind howls down to a point that you can see on the ground within range.
Ok, this last one doesn't contain the phrase "strong wind", but it's a whirlwind! I think it counts...
Green Layer
This layer requires "A passwall spell, or another spell of equal or greater level that can open a portal on a solid surface". Unfortunately, the only other portal related spell I can think of that uses a surface is teleportation circle, which specifies "the ground" (PHB, pg. 282):
As you cast the spell, you draw a 10-foot diameter cricle on the ground inscribed with sigils that link your location to a permanent teleportation circle of your choice...
I imagine the intent of this wall's requirement is to cast the spell similar to passwall on the wall itself, not on the floor next to it, so I don't think teleportation circle can help us.
There is demiplane (thanks @Nick), which says (PHB, p. 231):
You create a shadowy door on a flat solid surface that you can see within range. The door is large enough to allow Medium creatures to pass through unhindered.
I think this would count as creating a spell of greater level that can open a portal on a solid surface, so demiplane should bypass the green layer.
(Thanks to @illustro for suggesting the next three spells)
Mordenkainen's magnificent mansion could be an option, since it somewhat ambigiously says (PHB, pg. 261):
You conjure an extradimensional dwelling in range that lasts for the duration. You choose where its one entrance is located.
Gate says a similar thing (PHB, pg. 244):
You conjure a portal linking an unoccupied space you can see within range to a precise location on a different plane of existence. The portal is a circular opening, which you can make 5 to 20 feet in diameter. You can orient the portal in any direction you choose.
These entrances or portals could be on the green wall (gate less obviously, but it could be oriented to be against the wall), thus serving as an alternative to passwall, although they're not explicitly cast against the wall like passwall is, so it's a DM's call as to whether this would work.
Arcane gate could also be an option, given that you could align the free-floating portals with the wall (PHB, pg. 214):
The portals are two-dimensional rings equal to 10 feet in diameter, hovering perpendicular to the ground. A ring is visible only from one side (your choice), which is the side that functions as a portal. [...] On your turn, you can rotate the rings as a bonus action.
The fact that it's hovering, it could be argued that this is therefore definitely not interacting with the wall, but the mentioning of rotation could imply that it could be aligned with the wall, as with gate.
Another suggestion (thanks to @CapnZapp) could be using the Nolzur's Marvelous Pigments (DMG, pg. 183), a very rare magic item that can:
allow you to create three-dimensional objects by painting them in two dimensions.
...
Thus, painting a door on a wall creates an actual door that can be opened to whatever is beyond.
I, as DM, would personally allow this as it seems similar enough to passwall for the purposes of taking down this layer, although it must be said that RAW this magic item does not cast passwall, nor does it cast "another spell of equal or greater level", so strictly RAW this one probably wouldn't work.
Overall, many of these options would be up to the DM, since passwall and demiplane seem to be the only spells that explicitly say that it's cast on a solid surface.
Indigo Layer
This layer requires "bright light shed by a daylight spell or a similar spell of equal or higher level." Those that spring to mind (or @illustro's mind) are dawn, wall of light, sunbeam and sunburst, all of which create bright light, although I should mention that clearly the light cantrip will not work (even though that's obvious) since it's not of equal or higher level to daylight.
From dawn (XGtE, pg. 153-154):
The light of dawn shines down on a location you specify within range. Until the spell ends, a 30-foot-radius, 40-foot-high cylinder of bright light glimmers there. This light is sunlight.
From wall of light (XGtE, pg. 170):
A shimmering wall of bright light appears at a point you choose within range.
From sunbeam (PHB, pg. 279):
A beam of brilliant light flashes out from your hand in a 5-foot-wide, 60-foot-long line.
From sunburst (PHB, pg. 279):
Brilliant sunlight flashes ub a 60-foot radius centered on a point you choose within range.
Also, if we include magic items, it's worth mentioning that the Driftglobe can cast the daylight spell, and that there's a magic item called the Gem of Brightness, which says (DMG, pg. 171):
The second command word expends 1 charge and causes the gem to fire a brilliant beam of light at one creature you can see within 60 feet of you.
Unfortunately, this won't help us since it wants to target a creature, and we need it to take down a wall. However, the next bullet point says:
The third command word expends 5 charges and causes the gem to flare with blinding light in a 30-foot cone originating from it.
The only trouble with the Gem of Brightness is that whether the effects of an uncommon magic item are considered "of equal or higher level" to a daylight spell, and the fact that the description of this layer of the prismatic wall specifically says "or a similar spell", and the effect of a magic item is not a spell (although a DM may allow it anyway, since it's light from a magical source with a similar effect to daylight; I'd allow it).
Driftglobe is probably alright RAW because it mentions the daylight spell specifically.
Violet Layer
"This layer is destroyed by a dispel magic spell or a similar spell of equal or higher level that can end spells and magical effects." Besides dispel magic, what else can "end spells or magic effects" in such a way?
Possibly greater resortation, since it can remove certain conditions like petrified and the like, but I'm not sure if it is similar enough to dispel magic, since it just stop a creature being affected by such spells or effects. This one might have to be a DM's call as well.
On the other hand, remove curse should be valid RAW, as it is considered to end curse spells; from bestow curse (PHB, pg. 218):
A remove curse spell ends this effect.
Hex also says this about it (PHB, pg. 251):
A remove curse cast on the target ends this spell early.
So remove curse is called out as being able to end spells early at least twice, meeting the criteria for being a spell "that can end spells and magical effects".
It's also worth pointing out that an antimagic field wouldn't work, since prismatic wall specifically says (PHB, pg. 269):
The wall can be destroyed, also one layer at a time, in order from red to violet, by means specific to each layer. Once a layer is destroyed, it remains so for the duration of the spell. Antimagic field has no effect on it, and dispel magic can affect only the violet layer.
Best Answer
It's debatable whether they would be affected at all
There are numerous spells that do damage or otherwise inconvenience a creature when that creature moves, or is moved, into their area of effect. Since you used Wall of Fire as an example, let's examine the text that spell uses:
This movement does not need to be voluntary: if you are shoved into a Wall of Fire, or walk into it yourself, you still "enter" the wall. However, note the similar (but not identical) text of Prismatic Wall (PHB, p. 269, bold added):
This "attempts" is crucial. If a creature is grappled and moved against its will, it did not "attempt" to move in the direction it is moving: it had no choice in the matter. For a creature to "attempt" to pass through the wall, it must be moving through it (or reaching into it) voluntarily.
This is also complicated by the fact that the second sentence in the quote above doesn't require intent, but simply that a creature "passes through" or "reaches through" a layer of the wall. It is a DM's prerogative to decide which of these sentences takes precedence in this situation.
What happens if you're pushed/grappled into the wall?
It's very hard to say. Strangely, none of the layers of the wall prevents a creature from physically passing through it while they are active (layers prevent things like magical and nonmagical ranged attacks from penetrating the wall). Nor is there any text that directly states that the wall is solid. It is described as "opaque", but that simply means you cannot see through it.
It would be a DM's call what would happen if you shoved a creature towards the wall, or attempted to run through the wall with a creature grappled. A case could be made for the idea that the wall is solid when you are thrown into it, because of the "attempts" mentioned mentioned earlier: this time, "attempts" is significant not because it indicates willing movement, but because it suggests a possibility of failure. If you "attempt" to move through something, it does somewhat imply that moving through it is not guaranteed, which might imply some level of solidity. Again though, that's very much a DM's call.
One thing seems clear to me: it does not seem likely that the wall is meant to be both harmless and intangible to creatures who are pushed through it involuntarily. If that were the case, a group of enemies could simply shove each other through it and avoid its effects altogether (except the unlucky one who is last).
What about the Blindness Condition?
(Credit to HellSaint for mentioning this in a comment). Passing into or through the wall is not the only way the wall could affect a creature (PHB, p. 267-269 [sic]).
This effect is one place where the rules seem somewhat clearer. The text says "moves" rather than "is moved", but forced movement is still movement. Likewise, the text of Wall of Fire requiring a character "enter" the area is similar grammatically to the text here requiring the creature to "move." And crucially, there is no text in this section restricting this particular affect to happening only the first time that a creature moves into its area on a particular turn.
Although there is no designer answer to this question as yet, the Rules As Written (RAW) seem to indicate that you could drag a creature close to the wall (within 20 feet) and subject them to the saving throw to avoid blindness. And conceivably, if the creature succeeds, you could then drag them away from the wall, and towards it again all on the same turn, subjecting them to the same saving throw once again.
Of course, you'd need to spend a 9th level spell slot to do so.