Rock of Areopagos (Areopagitou):
Right at the foot of the Acropolis is a rock that offers a nice view of the city, is accessible day and night and is free of charge. Many youngsters and less young enjoy the view of Athens below with a drink and some music (all of which you have to bring along). You can get there easily with the stairs, but be careful once on top, some rocks are very slippery. It is ideal at sunset and after dark, when you can take in all the lights of the city.
How to get there?
Go a little beyond the ticket booth at the Acropolis entrance in direction Plaka (with the Acropolis itself being on your right), you will see some stairs going up the rock.