I don't know how you get that from the passage. The implication is that Peter would be executed. We don't know how old he was exactly. Early church history states that Peter was executed under Nero's instructions. You probably know that Nero blamed Christians for the terrible fires in Rome. The apostles in Jerusalem were the obvious targets for the authorities. I agree that Peter almost certainly never went to Rome. There is no evidence to suggest that he did. If he did, why did Paul write so comprehensively to the Roman church? There is certainly no suggestion in Acts that Peter went to Rome. He was apostle to the Jews, not Gentiles.
There was a significant Jewish population in Rome, as you read Paul's Epistle to the Romans you'll notice that in several places Paul seems focused on his Jewish Christian audience, for example in chapter 1 he talks about the "icky Gentile pagans" in such a way as to lull his readers into a kind of trap, which he springs on them at the beginning of Romans chapter 2,
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Therefore you have no excuse, O man, every one of you who judges. For in passing judgment on another you condemn yourself, because you, the judge, practice the very same things. We know that the judgment of God rightly falls on those who practice such things. Do you suppose, O man—you who judge those who practice such things and yet do them yourself—that you will escape the judgment of God? Or do you presume on the riches of his kindness and forbearance and patience, not knowing that God's kindness is meant to lead you to repentance? But because of your hard and impenitent heart you are storing up wrath for yourself on the day of wrath when God's righteous judgment will be revealed." - Romans 2:1-5.
Paul is establishing a larger foundation here, in which he talks about how both Jews and Gentiles are equally condemned as sinners under the Law. That's the whole point by the time we get to Romans 3 and Paul says,
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for all have sinned and fall short of the glory of God," - Romans 3:23
That's the Law, Paul has just brought the hammer of the Law to all men, both Jew and Gentile. For all have sinned, none is with excuse, nobody can escape the wrath of God.
But immediately follows that up with the Gospel,
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and are justified by his grace as a gift, through the redemption that is in Christ Jesus, 25 whom God put forward as a propitiation by his blood, to be received by faith." - Romans 3:24-25a
Remember, Paul gave us the thesis statement for this epistle back in Romans 1:16-17, the Gospel is the power of God to save all who believe, Jew and Gentile, for this Gospel reveals God's righteousness by which He makes us righteous through faith. The righteousness, Paul says later, "that is by faith"
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But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it— the righteousness of God through faith in Jesus Christ for all who believe." - Romans 3:21-22
If Peter's ministry focus was on his fellow Jews, then Rome makes a lot more sense than Antioch. Paul went to Antioch because of his ministry to the Gentiles, because Antioch was a predominantly Gentile city, and is why followers of Jesus are first called "Christianoi" here, probably because the Greeks heard "christos" and were entirely unaware of the Jewish context (that this was a translation of Meshiach, the Anointed promised by God as Redeemer for Israel and the nations)--so all this "christos" talk possibly led the people of Antioch to refer to the adherents of this strange new religion as "christos ones", "Christianoi" aka "those of christos"--where "christos" would have sounded like "oily one" to a Greek. Thus these "oily ones" or "people who talk about the oily one". As there is a frequent agreement that "Christian" was initially meant to be a pejorative, only later being adopted as a badge of honor (e.g. 1 Peter 4:16, Acts 26:28-29)
So that Peter would leave Antioch and go to somewhere with a larger Jewish population, like Rome, which had the highest concentration of Jewish persons anywhere in the ancient Diaspora, thus the highest number of Jewish people of any city outside Judea/Galilee of anywhere in the Empire, makes a great deal of sense.
History of the Jews in the Roman Empire - Wikipedia
A thriving Jewish community existed in the "Eternal City", even with the Tiberian and Claudian explulsions taken into account.
-CryptoLutheran
