Warning: SPOILERS for The Falcon And The Winter Soldier Episode 1 - "New World Order"
The Falcon and The Winter Soldier's Bucky Barnes (Sebastian Stan) ages differently from Captain America (Chris Evans) because they're not the exact same type of super-soldier. Alongside Sam Wilson/the Falcon (Anthony Mackie), Bucky steps into the spotlight in The Falcon and The Winter Soldier, and he feels like he has a very long lifetime of crimes he has to atone for.
While Super-Soldier Serum by Dr. Arnim Zola (Toby Jones). Even after Rogers, who was Captain America by that point, rescued the 107th, Bucky was unaware that he was enhanced by Hydra. Barnes survived a fall to his death but he was recaptured by Hydra, brainwashed, and turned into the Winter Soldier. As Hydra's top killer, the Winter Soldier performed numerous black ops assassinations, before he was mind-wiped and placed into cryostasis after each mission.
When Steve Rogers faced his best friend Bucky Barnes again in the 21st century, the Winter Soldier appeared to be a dark mirror image of Captain America. Like Barnes, Rogers was also cryogenically preserved for almost 70 years before he was reawakened, and they both appeared to be roughly the same age that they were when they last saw each other in 1945. Bucky was born on March 10, 1917, so he was already older than Steve, who was born on July 4, 1918, even before they became enhanced. Further, Captain America was frozen and preserved as he was in 1945 without thawing for nearly seven decades before he was revived. The Winter Soldier was pulled in and out of cryostasis by Hydra. So, Bucky was about 28 back in 1945 when he was turned into the Winter Soldier, and he also kept aging during all of the time he was out of cryo for missions.
There's also considerably more mental wear and tear on Bucky than Steve, which has aged Barnes. Even though Avengers: Endgame – has an extended lifetime of sins he feels he must atone for. Bucky's age is further complicated since he was a victim of Thanos' (Josh Brolin) snap so the five years he was dead can be subtracted from his physical aging, if not his chronological age.
Comparing Bucky to Steve is also confusing: Barnes appears to be in his early 30s in The Falcon and The Winter Soldier but the last time fans saw Steve Rogers in Avengers: Endgame, he was an old man. Steve used the Quantum Realm to travel back to 1945 so he could live a life with Peggy Carter (Hayley Atwell). When he reemerged in 2023, the years had caught up with and aged him, so it appears that while the Super-Soldier Serum offers super strength, endurance, and enhanced healing capabilities, it is not a fountain of youth or a key to immortality. Even Bucky's variant serum should allow him to age normally (unless he gets frozen again).
A side effect of the Super-Soldier Serum that the Winter Soldier shares with Captain America is that neither can get drunk. In The Falcon and The Winter Soldier's premiere, Bucky noticeably downs several bottles of beer while on a date with Leah (Miki Ishikawa) and he suffers no ill effects. This is because the Super-Soldier Serum makes the metabolism burn alcohol at a rate four times faster than a normal person. Ironically, Steve Rogers discovered this inconvenient truth when he was at a pub in London back in 1945 when he was mourning Bucky's death.