If we are considering "the universe" to mean the spacetime that we exist in, there could be an "outside," but we just don't know, and there's no indication of such an outside, or anything about what it would be like.

By way of infinite spacetime, yes, there is only a part of spacetime that we can observe, because the farthest part is moving away from us faster than the speed of light. I seem to recall there having been some estimations of how large all of spacetime is, observable and unobservable, and that it has a finite size.

That said, there does not appear to be a limit to the size of spacetime. Based on what is currently known, spacetime is expanding, the expansion is accelerating, and there is no limit to the expansion.