Many monasteries have their own water source. A place were you can fill your waterbottles.
Most sources have nice carved marble slabs, a place to sit and a metal drinking cup on a chain. In the Iviron source a doubleheaded eagle is carved in the marble.
The water comes out a lions head and the doubleheaded eagle is again present.
This is inside the guestquarters of the Russian monastery. Not only for drinking but there is also modern soapdispenser to wash your hands.
In the courtyard of Docheiariou monastery there is a water well with fresh water. According to the monk we spoke, it is a miracle that the well gives fresh water for it is standing very close to the (salty) shoreline.
Another water well is located in the courtyard of the Serbian monastery.
In the small streets of the town of Karyes here and there are some watersources. They can’t be all made of marble.
There is plenty of delicious water on Athos that can be safely drunk. I never got sick from it.