The prophet may be able to convince some people that he is the resurrected Jesus$^1$, but he won't have any conclusive proof.
The earliest Gospel in the New Testament, Mark, was written sometime after 66 C.E. by an anonymous author. That's at least 35 years after the apparent crucifixion of Jesus$^1$. No one who would have actually met Jesus$^1$ actually wrote anything about him. Worse, they probably didn't get a first hand account, as any contemporaries would have to be far older than the 25 year life expectancy of the time nor did they speak the language.
So, even if this person was the actual reincarnation of Jesus$^1$, he would probably not match the description given in all existing evidence. Though people might expect that, it certainly reduces the availability for producing evidence.
His only chance of providing evidence for his divinity would be through the use of miracles or very clever magic tricks. Or simply abandon evidence and turn on the charm.
$^1$Not The Jesus
There seems to be a clear misunderstanding of what life expectancy means. Or at least, it's a subject everyone thinks that there are the only ones who understand it.
It's not a maximum, this is obvious because the Jesus character lived to be 33. It's an average. It includes childhood deaths, which lower the average. I linked to the table, but I'll calculate some conclusions here, for those who don't wish to do the simple math. The apostles were younger than Jesus and at least one would have needed to live 35 years to write the gospel of Mark and a couple others would need to live up to an additional 30 years to write the remaining and final gospels. Let's see what it takes assuming the apostles were all 25 year old men.
These are the dates of writing along with the probability that, when the gospel was written, any specific 25 year old man in the Roman Empire contemporary with Jesus was still alive (just alive, not the author):
Mark was written around 68 C.E. - 13.5% probability any particular contemporary of Jesus was still alive.
Matthew was written around 70 C.E. - 13.5% probability any particular contemporary of Jesus was still alive.
For the entire population of 25 year old men that were contemporary with Jesus, 13.5% would survive to be 70 years old (40,201 of 100,000 make it to 25, of those, 5,432 make it to 70 years old. 5,432/40,201 = 13.5%).
Luke was written around 85 C.E. - 2.34% probability any particular contemporary of Jesus was still alive.
For the entire population of 25 year old men that were contemporary with Jesus, 2.34% would survive to be 80 years old (40,201 of 100,000 make it to 25, of those, 944 make it to 80 years old. 944/40,201 = 2.34%).
John was written around 90 C.E. - 0.55% probability any particular contemporary of Jesus was still alive.
For the entire population of 25 year old men that were contemporary with Jesus, 0.55% would survive to be 85 years old (40,201 of 100,000 make it to 25, of those, 225 make it to 85 years old. 944/40,201 = 0.55%).
This is excluding any correlations those specific people might have, for instance the fact that nearly all of the apostles were executed for what they were teaching or the significant amount of travelling they did.