I seriously doubt he was, but since the theme of the film was Rocky losing his edge, could it be that even Paulie was posing a threat to Rocky? It obviously parallels the scene in the first Rocky when Rocky subdued an angry, drunken Paulie when he was screaming at Adrian. But if Rocky can handle an angry, younger, drunk Paulie...why was he so fearful and defensive against a heavier, depressed, hungover Paulie?
I don't think he was fearful, but he was being defensive so that Pauly didn't hurt him. While it's not a straight up fight (Either street or boxing type match), a wild swinging, angry drunk can cause quite a bit of damage.
Even a trained fighter can be caught off guard by a wild attack, so I think that Rocky was just moving around and making sure Pauly didn't get in a lucky shot. If he gets dazed by some wild looping punch, then he's defenseless against an angry, out of control aggressor.
Also, while you could make the argument that it is foreshadowing the self confidence crisis to come, the scene that you are describing is just after the sports announcer talks about Rocky successfully defending his title for the 10th time, so at that time in the movie there really is no reason for him to be fearful at all.