We will be there on a port stop between 2-9 PM in November and will be the only ship in Patmos. Is it easy enough to catch a taxi to take us to the Caves of the Apocalypse and the Monastery of St. John?
you can take a taxi up to monastery & walk down... the Cave thing is on the path down (sort of a tourist trap) . I think it wold be worthwile to taxi to a nearby beach I hear they are lovely, didn't get there because my companions not interested, but since those cruises don't usually give you beach time, I'd take advantage. Take the taxi driver's card & phone him for pick-up.
There are buses from the port area up to the monastery too, and I would guess they schedule them around cruise ship arrivals. The bus stops at the Cave of the Apocalypse on the way back down and lets people off if you’d like. After seeing the cave, you can either catch the next bus or walk down to town.