Azula. He takes that name, memorises it, files it away in his mind in case it ever becomes relevant. "I will remember." The important aspect to remember is she is responsible for harming Aang at some point in his past, which means if Bucky should across her in a combat scenario she will be a target.

"I don't know." he answers truthfully. "We come back exactly as we were when first brought here. My hair was cut before the arena, now it is long again."

As much proof as was needed that it wasn't just them fixing up their corpses and shocking them back into life but an actual reset. Not to mention some people's corpses were surely damaged beyond possible repair, like anyone who had fallen into the gears on the escalators. It's a puzzle, one Bucky can't begin to figure out, it wasn't part of his skillset.