The mizzium apparatus allows an attuned creature to do the following:
While you are wearing the mizzium apparatus, you can use it as an arcane focus. In addition, you can attempt to cast a spell that you do not know or have prepared. The spell you choose must be on your class's spell list and of a level for which you have a spell slot, and you must provide the spell's components.
You expend a spell slot to cast the spell as normal, but before resolving it you must make an Intelligence (Arcana) check. The DC is 10 + twice the level of the spell slot you expend to cast the spell.
On a successful check, you cast the spell as normal, using your spell save DC and spellcasting ability modifier. On a failed check, you cast a different spell from the one you intended. Randomly determine the spell you cast by rolling on the table for the level of the spell slot you expended. If the slot is 6th level or higher, roll on the table for 5th-level spells.
If you try to cast a cantrip you don't know, the DC for the Intelligence (Arcana) check is 10, and on a failed check, there is no effect.
Emphasis mine. All of the spells one can roll for on the tables for a failed check have a casting time of 1 action (or longer—which still uses only your action). However, there is no stipulation that the casting time of the spell you attempt to cast must be 1 action. This leaves it unclear on how one should resolve an attempt to cast a spell with a casting time of 1 bonus action which fails. What if the caster has already used its action for the current turn? What happens if the attempted spell has a casting time of 1 reaction, like shield? Is there any reason why the item would not allow one to attempt to cast such a spell?
Note that I am looking for rules here (if any)—not rulings. An accepted answer should point to a set of rules which leads directly to its conclusions.
Best Answer
The intent appears to be that the random spell just comes into effect, at the cost of whatever the casting time of the chosen spell was.
There is no clear rules answer here. The best we can do is to try to suss out the intent based on a couple observations. I know you asked for rules, not rulings, but sometimes it just doesn't work out that way. The rules aren't perfect and cannot account for every possible interaction. sometimes rules-informed rulings are the best you can do.
1. There is no restriction on the casting time for the spell we attempt to cast.
The item description simply states no casting time restriction, so I should be able to attempt to cast a spell with a casting time of one bonus action, or a longer casting time. However -
2. Spells with a casting time of 1 Reaction are unclear, but probably shouldn't work.
Spell with a casting time of 1 Reaction can only be cast in response to the trigger stated in the spell description. So the apparatus definitely doesn't allow you to just cast shield whenever you want. It is not clear to me that reaction spells even work at all. The rules for reactions state:
The Apparatus does not state that it allows you take reactions, but the rules for reactions require that a feature allow you to take a reaction. Unfortunately, I have to go with a ruling here, instead of a clear rules based conclusion. Based on the above, I'm ruling that reaction spells are not eligible, but there is some room to rule otherwise.
3. Not every spell on the random tables has a casting time of 1 Action.
Rolling a 2 on the 4th level table yields conjure minor elementals, which has a casting time of 1 Minute. Now, normally it takes 1 minute to finish casting conjure minor elementals. It is not said that you have "cast" the spell until the minute is up. However, the Apparatus says "you cast a different spell from the one you intended." Based on this, if you fail the ability check, and roll conjure minor elementals, it is cast completely at that moment, rather than the casting beginning at that moment. Unfortunately, this is once again not entirely clear, so this is my ruling based on what is here.
Given this interaction with conjure minor elementals, I would extend it to bonus action spells. You attempt to cast a bonus action spell with your bonus action, you fail and roll on the table, and you get a different spell, still having expended your bonus action. This is the most intuitive ruling that results in not having to weird stuff with the action economy.
Medix2's answer comes to a different conclusion about shield and covers some similar interactions with other magic items that don't really make things much clearer, which just highlights the fact that there is no clear rules answer with this, and an informed ruling is the best you're going to get.