They don't do a great job of explaining it, but it's pretty clear the movie version of WS, like the comic version, has stats at Caps level.
Ok now this I now for certain is incorrect. In the comic it is specifically mentioned that after the Soviets found him they checked his blood for the serum, he didn't have any, and they didn't have any to give to him.
The WS is over 100 years old and still looks pretty good, he has had enhancements. It is not just the arm.
Winter Soldier mentions being 100 years old. He was besties with Cap in the 40's. You don't stay that young from pushups.
Looking young comes from having been frozen for many years.
edit: the most concrete information is that he was given unspecified "treatments", not sure who stated this in the movies. Not the super serum though, because they didn't have any. The combination of training, the arm, and whatever these "treatments" are was meant to pick him on the same level as Captain.
Here's a good post about it I found.
"Buckys case is very unique and interesting. But in most cases it can be broken down simply. In comics bucky is peak human and his metal arm solely gives him superhuman strength. In the MCU the dynamic is actually quite similar. He is enhanced to some extent but his metal arm is stronger than the rest of his body.
Unlike what many people say Bucky did not receive a recreation of the super soldier formula. In fact it's left very vague what happened to him that enhanced him. We just know that the experiments allowed him to survive the fall in the first movie at the cost of his arm. And afforded him some degree of superhuman strength and ability.
Shown here when he manages to evade a car and drop down from an incredible height without damage:
Aside from that buckys arm itself is incredibly strong even vastly stronger than both of caps arms. Shown here where buckys arm overpowered Steve with little to no difficulty
Aside from that Steve is incredibly strong too and was able to curl a helicopter.
So in summary, yes bucky’s physicality is similar to that of steve’s but his arm is greatly stronger."