In theory, if the universe is finite then it has finite dimensions that could be measured.
Of course the universe is so "large" and our intellect so "small" that "we" can't do it... at least not yet.
As to whether or not the universe is infinite in therms of "having always been here
, it is pretty doubtful that "we" will ever be able to prove it either way. Even if some "beginning" like the big bag were "proven", the question would remain "what was there before the big bang?"
In any case, these so called "thought experiments that people come up with are ultimately meaningless snd prove nothing.
For hundreds of years philosopher tried to prove the existence of god with so called logic, philosophy, "thought experiments", etc... and each of the "proofs" have been found to have flaws that boil down to: "if A = B, and I claim a bunch of other stuff that may or may not be true, then I can make a great leap away from logic and facts and claim that I just proved the existence of god". Thus far, however, the soft science if philosophy has not proven either god... or a finite universe.
The bottomless bucket "thought experiment" is right up there with dumb "proofs" that god doesn't exist like: "if god is all powerful, then can he make a something so big and heavy that even he can't move it?" At first blush it looks like a flaw in the "all powerful" thing, but in reality is a dumb meaningless statement that proves nothing.