No. Pope isn’t a rank. There are only three ranks of Holy Orders–deacon, priest, bishop. The Pope is the Bishop of Rome. Pope and pontiff describe the additional responsibilities–the Petrine Ministry as Pope Benedict calls it–that come with the office of Bishop of Rome.
Election as Bishop of Rome comes with it several added ecclesial responsibilities. The greatest among them is the Petrine Ministry – the office of Supreme Pontiff and Servant of the Servants of God. Like any other bishop, the Pope is the bishop of his diocese (Rome) and the pastor of his cathedral (he has an archpriest to run his cathedral like most bishops have a rector or a dean to run theirs). He is the metropolitan of the Roman Province, the primate of Italy, and the patriarch of the Western Church (a title that Benedict has left vacant).
However, in the end, the Pope is a bishop…the Bishop of Rome…before he is everything else. He’s a bishop with huge responsibilities beyond just his own diocese.