Cumulative chance does mean that the chance accumulates each time, yes. Each time you drink it after the first, there is a chance that it will age you instead of youth-en you. That chance is 10% initially, but increases by 10% each time.
So the first time you drink it, it has its normal effect. The second time, there's a 10% chance it has the opposite effect and ages you. The third time, a 20% chance of aging. The next time, a 30% chance; the next 40%. And so on.
This chance doesn't ever reset, it just accumulates.
(This does seem to be a pretty poor potion of longevity. The local Better Magical Business Bureau will probably need to open an investigation into deceptive advertising and naming practices.)
As for the wording, this isn't actually incorrect, perhaps just unfamiliar. A "cumulative chance" can be used as part of a statement of an observed or imposed rule or physical law, or it can be used as part of a sentence describing the current state of a cumulative chance. The normal usage is the former, with the latter being derivative of the former and more often seen in summaries or after-the-fact status reports and similar contexts. The default meaning of "cumulative chance" is describing how the chance accumulates, and is what should be assumed if in doubt.
I'm going to take a markedly-different approach from many of the others; hopefully this perspective is useful, too.
Start by looking at the nearest-neighbor spell to augury: aura of life. 30' radius, resistance to necrotic damage, regains 1hp at start of 0hp turn. Very easy, and the only thing really left up to interpretation is "nonhostile."
Most spells are like this: they're very clearly... spelled out, using defined terms. [/rimshot]
Augury is a different type of spell.
Augury is a super-vague spell, with lots of room for GM interpretation, preference, style, &c. And you're a first-time GM, so you feel like you've got little guidance from the spell. You're right, by the way.
There's a reason for that: while most spells are a manifestation of the player character's will, this spell is designed to do something fundamentally different. Augury allows the player character to engage in conversation with the GM. Not the player; the player character.
I want to be clear on this: the purpose of augury (and some other divinations) is markedly different than most other spells. Once you realize that, things fall into place.
At some tables it's perfectly appropriate for a GM to say to the players "hey guys, if you take on this dragon you're all probably going to die." If necessary, this is justified in-fiction as "your characters are noting the number of dried skeletons of previous adventurers, they know stories of Smaug from their childhood, they've trekked through miles of desolate wasteland just to get here." Or it's hand-waved away.
At other tables that would be completely unacceptable. It'd ruin immersion, it'd destroy people's roleplay, it'd cheapen the game. Crossing the boundary between in-world and at-table knowledge/conversation/interaction is forbidden.
Augury explicitly crosses that boundary, in a way supported in the fiction. Augury allows the character to talk (specify course of action taken very soon) and the GM to respond: "that should go well," "uh, I wouldn't do that," "kinda mixed bag," or "meh."
But what about timing, subject, and adjudicator?
My best advice is never to spend more than 30 seconds on an augury. Listen to the question/proposition, lean back, think for five seconds, and give an answer. If your players or PCs want better answers, they can wait (or pay) for divination, commune, or contact other plane.
Best Answer
"Cumulative" in this context means Additive
Cumulative is a word formed from shortening "Accumulative", which means to add together. So when the text reads "a cumulative 25% chance of failure per casting", it's intended to mean "accumulate 25 percentage points worth of chance per casting".
Because of this, it is correct to treat the math as being (X / 4) chance of failure (random result), where X is the number of times prior to the current casting of Augury that you've cast Augury. This means that on the first casting, there's a 0% chance of failure, then a 25% chance, then 50%, then 75%, and on the fifth casting onwards, the result is always random.
Either roll a single d4, like you suggested, using rolls of (1,2,3,4) as the minimum required for success for each successive roll, or use d% when making this roll, using (1, 26, 51, 76) as the minimum required rolls for success.
If the rules intended you to roll multiple dice for a multiplicative calculation, like your second option suggests, the rules would expressly instruct you to do that.