The office of the papacy isn't tied to the city of Rome due to any intrinsic quality Rome has that Jerusalem hasn't, it's tied to the apostolicity of the particular Church located there (a "particular Church" being distinct from a "church building" or a single ecclesial communion). Rome is the apostolic Church of the Apostle Peter, who set its foundation and was martyred there:

"How happy is its church, on which apostles poured forth all their doctrine along with their blood! Where Peter endures a passion like his Lord's!"

Tertullian, Prescription Against Heretics, Ch. 36

Rome wasn't the only city Peter preached in, but because Peter was martyred in Rome the bishop of this Church became his direct successor:

Peter appears to have preached in Pontus, Galatia, Bithynia, Cappadocia, and Asia to the Jews of the dispersion. And at last, having come to Rome, he was crucified head-downwards; for he had requested that he might suffer in this way... After the martyrdom of Paul and of Peter, Linus was the first to obtain the episcopate of the church at Rome. Paul mentions him, when writing to Timothy from Rome, in the salutation at the end of the epistle.

Eusebius, Church History, Book III

Peter's office takes significance not from Peter being a "successor" of Christ (as Christ could have no successor), but rather because he was Christ's vicar (which is why you'll hear the pope being referred to as the "vicar of Christ," the word vicar meaning deputy), given a particular office and authority by Christ, which his successors have inherited. It isn't tied to the land (Rome), it's tied to the man (Peter), who himself is tied to Christ (the Man) and not the land Christ walked (Jerusalem, et al).

The same reasoning applies to other Christian sects with popes residing outside of Jerusalem and the Holy Land - here, you're probably most specifically referring to the pope of the Coptic Orthodox Church (currently, Tawadros II), who presides over the Orthodox Church at Alexandria, which claims the apostle Mark as its founder.

It may surprise you to know that as a title, the word pope being applied to the Church at Alexandria is older than its application to the Roman Church! This doesn't have anything to do with the office of the Roman papacy as it's historically understood (the successor to Peter). The word itself is simply derived from Greek (and Latin), meaning "father," and was applied to many bishops in the early Church. It evolved to hold the special significance it does today in the West over the course of several centuries!

When the Christian church was first founded, organizational structure was fairly limited and the ultimate authority in the church were the apostles. Decisions were made at church councils, such as the Council of Jerusalem; and other than the apostles, church leaders from one church did not hold any authority over church leaders in other churches.

Beginning in the late 2nd century, some centres of Christian worship began to be recognized as more important than others, and the leaders in those communities began to be recognized as leaders over a regional area.

It was not until the end of the Roman persecution of the Christian church in 313, that the idea of regional authority became formalized, and "metropolitans" or "dioceses" were created which had authority over the entire province, and in some cases multiple provinces. The first three metropolitans were the bishops of Alexandria, Antioch, and Rome; these are also known as the "Petrine Sees." The centuries that followed saw the establishment of several metropolitans, roughly corresponding to the provinces of the Roman Empire; and it also saw these bishops gradually acquire more and more influence over church affairs within their province.

In 531, the Emperor Justinian began referring to the bishops of five certain metropolitan sees as "Patriarchs", with authority over entire regions. These five Patriarchs were the bishops of Rome, Constantinople, Alexandria, Antioch, and Jerusalem. While there was a hierarchy of patriarchs, they were essentially considered equal in authority and status to each other (although the patriarch of Rome considered themselves the principal patriarch, a claim not necessarily recognized by the other four).

As time went on, each patriarch continued to increase his authority. Simultaneously, wars and political instability divided the patriarchs, as they were no longer under one government but several. Rome was supported by the Holy Roman Empire of Charlesmagne and eventually grew to become the church of Western Europe, the Roman Catholic Church. Constantinople was supported by the Byzantine Empire and eventually grew to become the church of the East, the Orthodox Church. Alexandria found itself trapped in the Muslim world and forged its own existence as the Coptic Church. Antioch and Jerusalem found themselves in the middle of a battleground between all three worlds (Byzantium/Muslims/Crusaders) and struggled the most to start their own church, with Antioch leading the Syriac church and Jerusalem eventually falling under the influence of its neighbours.

Some of these patriarchs chose to adopt the term "pope" to signify their leadership over their entire church--for example Rome and Alexandria.

After the division of the church, the fortunes of each church generally matched the fortunes of the nations that supported them. The Coptic church was never supported, the Byzantine Empire collapsed which damaged the support of the Orthodox church, while the Roman Catholic church had the support of the nations of Western Europe, who would go on to conquer most of the rest of the world, while bringing missionaries along to convert the rest of the world to Roman Catholicism.

This is why the Pope lives in Rome--because he was originally the bishop of Rome, head of the church of Rome, one of five church leaders; and due to the events of world history, his church grew to dominate the world of Christianity just as the nations of western Europe grew to dominate the world through colonialism.