"Claim to fame" is an idiom. It is the reason, already established as fact, that something is "famous".
"Claim to fame" here means having right to the title "the world's oldest Christian monastery"
Когда вы претендуете на славу..... When you claim to be famous for being "the world's oldest Christian monastery"....this is not yet established fact only a pretension.
Yes, translation back and forth is difficult. It requires an incredible knowledge of both languages in order to do it correctly. That is why you see so many bad translations. I confess that I have been responsible for a few of them. :)
Sometimes, there is no translation, then you have to be author as well as translator.