There are two major entrances to the Acropolis: the West Entrance (which leads directly up to the Parthenon and its friends; and the Southeast entrance (which takes you through the “back door” of the South Slope of the Acropolis before putting you on the summit itself). You want this latter entrance.
Take the Metro to the Acropolis Station (just one stop south of Syntagma Square on Line 2). Head upstairs following the “Dionissiou Areopagitou” signs. Escalators will carry you up to the surface level. When you hit the sidewalks, you are on Makrigiani Street and should be headed north. Walk straight ahead about 20 yards. This will dump you onto Dionissiou Areopagitou Street. Just across that street and a little to your left, you’ll see a very small park-like area. That’s the entrance to the South Acropolis.
(Just east of this area, by the way, are public restrooms. You’ll find stairs leading down to them. You might consider using the facilities before you tackle the Acropolis.)