One possible older position is that of Samaritan High Priest. Although a longer period is claimed, it seems that the office goes back to the Hellenistic period.

I would also wonder about priesthoods in Ancient Egypt, but I'm afraid I have nothing to offer there.

In tradition, The Emperor of Japan is claimed to be traced back 2,600 years. The position still exists today as a figurehead.

http://en.wikipedia.org/wiki/Emperor_of_Japan#Origin

You could easily dispute much of the earlier history, though you could easily do the same for the history of the papacy.

Emperor Sujin is the first emperor who is generally agreed to have existed, though some pin the dates as late as the third or fourth century. If you take the latest date, you would roughly coincide with Roman Emperor Constantine and the Council of Nicaea, at which point the Christian church was taking on a more organized form.

A lot of this question is going to come down how you consider early Christian history. Was the Apostle Peter the first Pope? He is in Catholic tradition, but other Christian denominations would disagree, saying the office didn't really exist until much later.

Followup question, with the Pope being the bishop of Rome, might some of the other city bishops be older and continuous as well? Perhaps the Greek Orthodox Patriarch of Jerusalem or the Archbishop of Antioch?

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