If you want to be near the Acropolis and other attractions, stay in a lovely apartment and be helped by a great host, then stay here, simple. Everything within a few minutes walk. Athens is not very expensive generally speaking.
We took the Bus from Athens Airport (Rosana's advice) for 6 euros. Then a 20 minute walk to the Apartment (if you can walk past the lovely shops that is). Thank you Rosana. (see you again)