As far as I know, any adult male can go. I don't know what the procedure is these days, but it probably still includes going to the US Embassy in Thessaloniki to get a criminal background check which you take to the Ministry of Northern Greece which, if you find a way to communicate with them, will give you a three day pass which is all the longer a foreigner is allowed to stay.
Then you take an early morning bus to Ouranoupoli and board a boat. I don't know about now, but they used to take your passport and the three day pass on the boat, which was then transfered to the bus driver at the Athonite Port of Dafni, who then delivers it to the authorities in the village of Karyes, from whence you are given the real three day pass. If you want to stay, those three days must be used to find a skete which will vouch for you and help you get permission to stay longer.
Perhaps there is another way these days, this info is 20 years old.
It would probably be easier to find a monastery elsewhere in Greece, under whatever bishop you choose, and go there.
As far as I know everyone speaks modern Greek. The services are read in Koine though, a.f.a.i.k., (someone will correct me if I am wrong).