Bucky Barnes does not have superhuman abilities.
But neither does Steve Rogers. Instead, Steve is simply at peak physical fitness for a human being (to the point where he may appear to be superhuman). It would seem that after Captain America: The Winter Soldier, Bucky also shares these traits.
In Captain America: The First Avenger, Steve Rogers (Captain America) rescues Bucky Barnes (The Winter Soldier) from Hydra captivity, finding him having recently been experimented on by Dr Arnim Zola.
It is theorised that Dr Arnim Zola was working on his own version of Dr Abraham Erskine's Super Soldier Serum, and it is this that was being tested on Bucky when Steve found him.
The fact that;
Bucky survives his fall into the ice in Captain America: The First Avenger before being brainwashed by Hydra and turned into The Winter Soldier
Implies that Bucky did receive some kind of Super Solider serum from Dr Arnim Zola, explaining how he can go toe-to-toe with Captain America.
This theory is further solidified in Captain America: Civil War, when we learn that;
there are 5 other "Winter Soldiers" that worked for Hydra, each of whom had received a version of the Super Soldier Serum that was created by Howard Stark prior to his death.