No, Rocky is not superhuman, and there are numerous explanations for how he achieves the seemingly impossible and beats Drago. A practical explanation is simply that Rocky can take tremendous amounts of punishment without being defeated. Throughout Rocky, Rocky II and Rocky III, it is emphasized time and again that one of his greatest assets in the ring is his ability to take a severe beating, and still be able to fight. In Rocky, there is the famous scene where Apollo is sure he has won, even Mick thinks Rocky is done for, but Rocky is somehow able to get back to his feet and continue the fight. In Rocky II, Duke points out to Apollo that he saw Rocky get beaten as bad as any man he's ever seen, but nothing would keep him down. In Rocky III, Mick says that the beating Rocky took from Apollo in Rocky II should have killed him, but it didn't. Then at the end of Rocky III, Rocky dares Clubber Lang to punch him as hard as he can as often as he can, so as to tire him out, because he knows he can withstand Lang's power. As Drago himself says in Rocky IV, "He's not human, he's like a piece of iron". This is also emphasized continually throughout Rocky Balboa; for example, one of the commentators mentions that Rocky now has a restaurant, and he speculates that the special might be "pounded chicken". So Rocky simply has an above average capacity for taking a beating without giving up. As he himself says in Rocky Balboa, "It's not about how hard you can hit, it's about how hard you can get hit and keep on going." Rocky can get hit very, very hard and keep on going!
Another part of the practical explanation for how Rocky survives Drago's punch is that Drago would not be able to punch him that hard in active competition. The strongest punch Drago registers is 2,150 PSI. This is more than a shotgun has at point-blank range, and were he to actually punch Rocky that hard, he would literally kill him. However, Drago registers that punch in a sterile, non-competitive environment against a stationary, undefended target; in an actual match, he would never be able to punch with that degree of severity and precision. So the answer to the question of how Rocky is able to survive such a massive punch is that he is never punched that hard. The reason for including the PSI readout during Drago's training is to increase the dramatic tension of the film -- the writers are essentially tricking their audience into thinking that Drago is more powerful than he seems.
Even taking this into consideration, however, it would posit Rocky is capable of taking a significant portion of a shotgun blast to the face hundreds of times over 15 rounds, which pushes the bounds of realism to an extreme. As such, a more symbolic explanation of how Rocky defeats Drago, despite Drago's seemingly overwhelming strength advantage, is tied up in the theme of the film. Rocky IV is largely a metaphor for the tensions between America and Russia which existed in the Cold War. Stallone himself has stated (in the famous Ain't It Cool public Q&A; see here) that the film isn't one to be viewed entirely literally, that is not exclusively about revenge or boxing, but rather an examination of the opposing sides in the war. The seemingly invincible Ivan Drago obviously symbolizes the Soviet Union, and his line "I must break you", perfectly represents his nation's sentiment toward America (or at least America's notion of the Soviets' intention towards them). During the fight Drago is far superior to Rocky in terms of firepower and strength, just as the Soviet Union was over America. However, Rocky's seemingly endless stamina and inability to lose is an obvious symbol for American national pride and heart, and the nation's unwillingness to be bullied or intimidated by a superior force. Rocky defeats the superior fighter because he has more heart; an obvious metaphor for the possible outcome of the War. In addition, Stallone has suggested that he intended the film as a message to the Russian people that there is more to life than military might, and that there could be a magnanimous solution to the tensions behind the Cold War. This is represented by the fact that not only do the initially hostile crowd begin to cheer for Rocky, but even Drago himself comes to respect him by the end of the fight. All of this depends upon a symbolic reading of the film, where the fight between Drago and Rocky is not so much a depiction of a real conflict which the audience is supposed to take literally, but a metaphorical examination of a real situation in the world at large.