Catholics do not hold to a theology in which there is an allotted sequence of time (such as a 'seven year tribulation' or a 'one thousand year millennial reign of Christ' after the said 'seven year tribulation') for souls to either believe or not believe in the return of Christ. Pope John Paul II referred to such ideas as 'millenarian fantasies', bereft of both common sense and susceptible to biblical cherry-picking. This would mean that the the pope's belief in the return of Christ would have little to do with the pope himself and more to do with the nature of Christ's return.
... and then shall all tribes of the earth mourn: and they shall see the Son of man coming in the clouds of heaven with much power and majesty. And he shall send his angels with a trumpet, and a great voice: and they shall gather together his elect from the four winds, from the farthest parts of the heavens to the utmost bounds of them.
It seems clear that Christ's second coming fulfills the other half of Jewish prophecies that suggest Christ comes in great power and as a king, in which all nations believe. This suggests that 'belief' in Christ, a matter dependent upon the Church and subjective reasoning, shall be universal since Christ is in his power and glory present to the world at this time. This power and kingship was revealed to a select few in Christ's resurrection appearances. It is upon these appearances and testimonies that the Church is founded in faith. The Church typically thus holds that the 'one thousand millennial reign of Christ' is occurring presently in the Church itself. The number, in this case, is not being taken literally, but figuratively (as much of Revelations is typically interpreted by the Church).
Regardless of these points, it is generally understood that the nature of Christ's coming is to gather the elect and to prepare the world for its final judgement. What exactly this means for the structure of the Church as we know it we cannot say. But we can say what shall vaguely happen to the general population that the Church just happens to be a part of. The population shall endure a final judgement, which is prepared by the second coming of Christ. We are given a few verses that indicate the nature of the second coming and how it relates to the final judgement:
He answered and said unto them, He that sows the good seed is the Son of man; the field is the world; the good seed are the children of the kingdom; but the tares are the children of the wicked one; the enemy that sowed them is the devil; the harvest is the end of the world; and the reapers are the angels. As therefore the tares are gathered and burned in the fire; so shall it be in the end of this world.
But when the Son of Man comes in His glory, and all the angels with Him, then He will sit on His glorious throne. "All the nations will be gathered before Him; and He will separate them from one another, as the shepherd separates the sheep from the goats; and He will put the sheep on His right, and the goats on the left.
It is written: 'As surely as I live,' says the Lord, 'every knee will bow before me; every tongue will acknowledge God.'
There seems to be judgement due to deeds that are already done, in which there is no opportunity for further justification (hence the final judgement), and judgement due to souls that have little ability to not believe in the greatness of God (though this of course is not justifying belief). Thus, in terms of what happens to a specific man (which the pope is), we are not granted the opportunity to understand, especially since that said pope is not even a real person yet. To assume we could know the judgement God has in mind for a hypothetical person is rather silly, since the Church firmly states that the judgement of a soul's internal and eternal state is not necessarily in our power to discern about any real person.
We can conclude that there are probably a limited amount of possibilities for such a man in this judgement: he can be of the elect or he cannot be. But those two possibilities are about as in depth as we are going to get in terms of knowing what's coming down the pike for a specific man. Although there might be speculation regarding what happens to the office of pope, I think such speculations are rather unfounded and bordering on being entirely irrelevant to what we actually know. This is not to mention how vague and nearly nonexistent the specifics of how Christ shall prepare the world for the last judgement are. Thus, talking about ideas and people in the context of their relationship to the coming of Christ can only be vaguely understood in terms of the general final judgement; specifics and further conversations veer towards foolish banter. In short, the most we can know about the papal office in terms of Christ's second coming, and hence in terms of the last judgement, is that the specific pope, just like the rest of us, shall either see his name written in the book of life or find himself bound to hell.