Steve must have lost more blood than he thought, or else he had been in shock and now that the adrenaline was wearing off it had left him exhausted. Or even maybe some of the stuff they had given Steve at the hospital was a soporific, but it was obvious that he wasn't in any state to be going anywhere.

"Shut up, idiot, just go to sleep."

Bucky can handle one night of having someone else in his place, he wants to prove to himself that he'll be able to have Steve as a roommate. He wouldn't wake him in an hour, he'd just get a blanket and cover him over, and leave him until he woke up naturally.