Spiros' place has a truly amazing view of the Acropolis - even better in real life than in the pictures. The outside terrace is first rate, the inside calm and cool. There is a good local place on the corner opposite where we enjoyed supper and Spiros had lots of other tips.
Enjoy (and as you run up the stairs to the top remember that that is the small price for an amazing view - and in any case the stairs are your gym.
we would definitely come back next time we're in Athens - the only downside being that you don't want to leave the flat at all as the best view is from the terrace !