Actually according to Irenaeus, Peter and Paul founded the church of Rome, not Peter alone. However, if you read the first chapter of Paul's letter to the Romans, neither Peter nor Paul founded the church of Rome. It was functioning just fine without an apostolic visit of any kind when Paul wrote his letter to the Romans in 56 AD.
But also: Do you remember my question: When did Peter become Bishop of Rome as legend has it?
Read your quote from Irenaeus carefully. Irenaeus says that Linus was the first Bishop of Rome, followed by Clement then Anacletus. So according to Irenaeus Peter was never the Bishop of Rome and there is no indication that he resided there for any period of time.
Well, I did read my extract from Irenaeus carefully. And it said:
The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric.
So, according to the extract from Irenaeus, the apostles committed the Church into the hands of Linus, then Anacletus, and then to Clement. So I am not sure where you get from this that Peter was never the bishop of Rome.
Now Irenaeus does not give indications of dates, but Eusebius states:
6. But this did not last long. For immediately, during the reign of Claudius, the all-good and gracious Providence, which watches over all things, led Peter, that strongest and greatest of the apostles, and the one who on account of his virtue was the speaker for all the others, to Rome against this great corrupter of life. He like a noble commander of God, clad in divine armor, carried the costly merchandise of the light of the understanding from the East to those who dwelt in the West, proclaiming the light itself, and the word which brings salvation to souls, and preaching the kingdom of heaven.Eusebius, Church History, Book II, 14.6
The reign of Claudius was from 41-54 AD, so I'd say that Peter arrived in Rome somewhere in that time frame. (By the way, the Acts of Peter and Paul corroborates the relative time frame). Hippolytus, Refutation of all Heresies, Book VI, Chapter 15, also provides corroboration.
So by all indications, it appears that Peter first arrived there and then was joined by Paul. As we all know, Paul arrived during the reign of Nero.
But when Peter is Bishop of Rome, they probably didn't do all that stuff. It was probably, "Okay, here's the head of the Apostles. I guess he's the leader of the congregation, I mean, anybody else want to lead while Kephas is here?" (Silence.) "Well, okay, now where were we? Oh yeah, can anybody put Kephas up for the night or for the next few weeks?"
It's looking back that we see that what he was doing there (stipulating that he was there and all) was bishopping, and to a certain extent proto-poping. So the evidence isn't going to be all crisp and cut and dried and filed in triplicate with a tape backup. It's anachronistic to expect it to be so, and conclusions drawn from the absence of such evidence will probably be unreliable.
And while he was apostle and person in charge, I wonder how many people would have been going, "Gee, how are we going to record this. IS he Bishop or are is being a bishop kind of included in being an Apostle?" It's hard to imagine that record keeping and nomenclature was high on their to-do list when Claudius was breathing threats and slaughter ... So, yeah Peter is bishopping and poping in Rome, but they hardle have the words for it and so it's easy to slip into saying Linus was the "first" bishop, because "Everybody knows" that before him that function was taken care of by Kephas.
We can't conclude much from the absence of evidence that probably would be absent or equivocal in any event.
"It is widely held that Peter paid a first visit to Rome after he had been miraculously liberated from the prison in Jerusalem; that, by "another place", Luke meant Rome, but omitted the name for special reasons. It is not impossible that Peter made a missionary journey to Rome about this time (after 42 A.D.), but such a journey cannot be established with certainty. At any rate, we cannot appeal in support of this theory to the chronological notices in Eusebius and Jerome, since, although these notices extend back to the chronicles of the third century, they are not old traditions, but the result of calculations on the basis of episcopal lists. Into the Roman list of bishops dating from the second century, there was introduced in the third century (as we learn from Eusebius and the "Chronograph of 354") the notice of a twenty-five years' pontificate for St. Peter, but we are unable to trace its origin. This entry consequently affords no ground for the hypothesis of a first visit by St. Peter to Rome after his liberation from prison (about 42). We can therefore admit only the possibility of such an early visit to the capital." http://www.newadvent.org/cathen/11744a.htm {Peter, Prince of the Apostles - Catholic Encylopedia}
In otherwords, Eusebius made it up. He had no evidence for his claims about a 25 year Roman Petrine Episcopacy. He didn't cite Irenaeus or anybody because there was no one to cite for his claim.
And Irenaeus never refers to Peter or Paul as bishops, does he? And he says that Linus was the first bishop of Rome --- appointed not by Peter alone, but by Peter and Paul, according to him. Notice that it is Peter and Paul, the two apostles that worked together there, per Irenaeus. The only time Paul could have been doing any such thing in Rome is a short window between his imprisonments there 62 - 64 AD.
So Irenaeus' story does not jive with what is later claimed about Peter.
It's possible that Irenaeus got his misinformation about them both being there doing this establishing together from the Acts of Peter and Paul, that you mention, which was a spurious work of fiction that no one put their name on.