if the universe was infinite it doesn't have this problem

Eh, not really. You're still bounded by the finite cosmological horizon. Unless of course you have access to super-luminal travel.

If I remember correctly you can take the power set of an infinite set and get a set with larger cardinality

Exactly.

So will I be stuck in some arbitrary cardinality?

It depends. If you use "subsets" as a generative ontological procedure, you would still be stuck by the finite time of the operation. If you consider "subset" instead as a conceptual relation, not some concrete process, you're not stuck in any cardinal.

Are the number of cardinality countable?

No. Once you postulate a countable cardinal, you get for free ordinals like "omega plus one", "omega plus two", etc. And since uncountable cardinals are ordered by ordinals, you also get for free more than omega uncountable cardinals.

Inaccessible is the next quantity for which you need a new axiom. Indeed, "inaccessible" is the quantity of cardinals generated in the process above.