Someone asked me recently, "Peter was in Rome?" Yes, reasonable tradition suggests that Peter was crucified in Rome under the emperor Nero. However, even Roman Catholic Bible scholars will tell you that it is not at all likely that Peter founded the church at Rome.
Brown and Meier, two of the best RCC Bible scholars of all time, suggest that the kernel of truth behind the tradition may be that the Roman church was a little more "conservative," more sympathetic to Peter's positions, than to Paul.
But I don't think it's appropriate even to translate the word episkopos as "bishop" in the time of the New Testament. I believe that, for the most part, house churches were run by a council of elders and that these elders were also called the "overseers" of the assembly (=church). So there was probably not even a single overseer (episkopos) of an individual house church, let alone one over a whole city. That doesn't happen until around the beginning of second century. That is not to say that there wasn't often someone more or less running the show.
So who was the first bishop of Rome? Linus may very well have been an overseer in Rome. He's traditionally second to Peter. Was he an overseer of all the churches in the city? We just don't know if it got that far that early. I personally doubt it.
Perhaps by the time of Clement, in the late first century, the role of overseer had developed enough for him to be called the overseer of the city. Even here, though, he doesn't talk of the leadership of Corinth in such "one man" terms. Ignatius in the early 100s is arguing for the authority of a single overseer, so even then the idea of a single leader of a city is still being solidified.
So Peter was not the first bishop of Rome, let alone the first "Pope," a concept that would not develop for several more centuries. Even in 1054, half the church denied that the bishop of Rome was of higher authority than the other bishops of key cities. Thus the Great Schism between East and West.