Question: Why does Bucky rip his jacket sleeve off before jumping out of the plane? Just for the sake of an audience reveal to demonstrate his artificial arm?
Answer: Aside from visual aesthetics and it being nice to see the metal arm. In world/practically I think (and I believe this is the reason why he tends to not have a left sleeve in most action we've seen him in prior films) it's also to assist with mobility during fights. The metal arm likely behaves a little differently than a human/bone and flesh arm does - and restricting that in a sleeve during a fight (especially in a thicker jacket sleeve) might mess with his reflexes. So removing the sleeve probably assists with the mobility of that arm.
Answer: That's the impression I got. Plus, he was expecting to go into battle, so perhaps he thought it would make him look more intimidating.
Chosen answer: The sleeve probably would have been ripped to shreds in battle anyway. He was just saving time by ripping it himself. Also, gunfighters would take off their coats in a duel or an arm wrestler rolling up his sleeve.