It's been 17 years since my wife and I were in Athens but the area behind the Acropolis is wonderful. The area is called the Plaka and is a small area of narrow alleys and lanes that hold everything imaginable for sale. I remember stumbling across a Gyro shop that was only 36 inches wide. Only sold food through a window and the food was fantastic.
I would like to tell you more places in Athens but we were on our honeymoon and spent most our of time at the beach or at the hotel. Plus we were in the military and not the most welcome folks.