Great question — answering as the UphillTech team:
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...
Full answer by Kevin Flynn: https://movies.stackexchange.com/a/18401 (CC BY-SA 4.0).
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