Koukaki is the quarter that lives in the Acropolis's shadow and barely notices it — residential, low-rise, café-quiet, the place Athenians actually have breakfast. Pair it with the museum that finally gave the south slope its anchor, and you have the calmest way to do the most-walked corner of the city. Go in the morning, before the boulevard fills.
Start on Odos Drakou, the pedestrian café street, and the small Plateia Koukaki above it — ordinary Athens, doing nothing in particular. Then the turn toward the grand: Tschumi's Acropolis Museum, glass floor over the dig, top gallery facing the Parthenon itself through the windows. Book ahead and arrive early; the light through that glass is the point.
Out onto Dionysiou Areopagitou, the car-free marble boulevard under the slope, past the Theatre of Dionysus where tragedy began — you can see it from the railing without a ticket. End at Akropoli station among the Makrygianni cafés, frappé in hand, the rock still above you.