Propileon (Προπύλαιον)
I had a very very mild iced coffee. Although they really want to call it something like 'café freddo' if I remember correctly. I don't speak much Greek, and they didn't seem to speak much English or my words weren't quite what they were used to, but we eventually managed to agree on what I wanted (cold coffee).
It's a café in front of a tourist hotspot, so I didn't really expect the best coffee of my life. There are really no more venues in the area anyway, and I was very thirsty, after walking all the way up the hill, so it was worth a try.
I got a table outside, looking onto the castle door, so it was a nice view.
I guess you could work inside, but it seemed quite dark. Not sure about wi-fi or power sockets though.