Do you believe him?

I'm of the impression that any way this problem is approached, the given solution is wrong. Any way you look at it, you can believe the archaeologist. If the "dating" means the coin is stamped, it could be a Roman relic and simply stamped as an approximate date sometime in the recent past.

1) If we assume that the coins were somehow how marked or stamped with the date, the problem becomes somewhat less intuitive but it's believable. It is possible that gold coins were marked sometime after the dating system was implemented. The problem doesn't specify if the coins were buried, in a building, or whatever. If the coins were stamped specifically with a date of 64 B.C. and buried with soil, fossils and stria from around that time period, then it becomes less and less likely but it is still possible. It may not be logical, but it's certainly possible. We've already found plenty of disturbances and irregularities in stria, so even a disinterred and reburied coin could be found with things of a different era.

2) The second possibility is that the coins were marked or stamped with 64 B.C., but that marking bore no relation to the date or the dating system. I can engrave something on a coin now, bury it, and in a few thousand years somebody might speculate about what it means. Perhaps a new dating system might be implemented in the future, and my random markings will coincide with that. Again, it's not logical, but it's certainly possible.

3) In archaeological terms, "dating" means radiocarbon dating of the age of the object. Bones dated from the distant past aren't stamped with the date. So, to an archaeologist saying a coin is dated 64 B.C. means the coin originated around that time. There need be no physical date stamped on it. The only real suspect here is the accuracy of carbon dating.