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Bucky was friend of Steve Rogers before Rogers became Captain America. The movie Captain America: The Winter Soldier shows events 70+ years after that transformation. Captain America is so young because he was frozen for 70 years, and being a super soldier helped him survive (normal humans can't survive under frozen conditions).
Why hasn't Bucky aged like Peggy? If Zola needed to survive using cyberspace, it means HYDRA doesn't have the tech necessary to stop ageing processes.
Because Bucky was also frozen for most of that time:
Dialogue from The Winter Soldier makes it clear that Hydra put Bucky back into suspended animation after completing each mission, and only unfroze him when they needed him:
Zola: The procedure has already started. You are to be the new fist of HYDRA. Put him on ice.
[...]
Hydra Scientist: He's been out of cryo freeze too long.
Captain America: The Winter Soldier (2014)
Zola's line also indicates that Hydra did something to Bucky, which may have been similar to the procedure performed on Rogers and may have helped him survive the cryo process. Dr R Dizzle talks about that more in an answer elsewhere
