First of all, he is not in the hands of the FBI yet, he was just detained by the Greek police, though it seems that he was arrested specifically by the request from the US authorities. Further, this dude might be a Russian citizen so the US authorities could encounter some issues extraditing him to the US where he is to be tried. In fact, Greece may have to make a tough choice right now since Greek authorities seem to be slightly pro-Russian these days but the country is still a NATO member anyway. Things are likely to get quite complicated here
Why do you think that Greece is pro-Russian these days? You maybe mixing Greece with Turkey?
It is not about just these days
Greece is an Eastern Orthodox Church country like Russia, and Russia in the past always supported or even directly participated in the warfare at the Greek side against same Turkey. Now Greece is one of the few older NATO members extensively using Russian weaponry, so there is substantial Russian economic presence in the country. Regarding Turkey specifically, Russia could very well play a role of an intermediary or mediator between these two countries. In any case, Turkey has never been Russia's ally and will never be
We don't have many Russian made weapons. Some APCs, a couple hovercrafts and one s300 system. Cyprus is the one that the military mostly consists of Russian weaponry. Greece in 2005 wanted to turn to Russia but was actively blocked by the US (google "Pythia 1 plan"). Greece always was a yes man of the west and always will be. Now the actual public opinion? that's different.