Godel doesn't say that there is any particular question about reality that we cannot answer

Of course! It's a theorem about math. There are no theorems about reality.

Am I correct in thinking that this rules out the possibility of a GUT, at least if a GUT is defined as a model that answers all questions.

Yes and no. You can build computers that enumerate proofs even in universes with simple and known physics, like the Game of Life. But to mathematically define something like an infinite Game of Life grid, you need integers, and we don't have a complete axiomatization of those. So you could have a GUT that's completely defined "relative to the integers". I guess most physicists would accept that as a good enough GUT, even though it's incomplete in the Godelian sense.