The Acropolis is the one walk in Athens that needs no defending: a limestone rock crowned with the building every other building has been measured against since. But the rock is only half of it. Do the slopes first and the summit second, and the climb earns its ending.
Start on the south slope at the Odeon of Herodes Atticus, the Roman theatre still hosting summer concerts, then walk on to the Theatre of Dionysus, where tragedy was invented in the open air. Both read well from the marble Dionysiou Areopagitou promenade — no ticket needed to see them from above. Then climb through the Propylaia, the great gateway, to the rock itself. Circle the Parthenon rather than confronting it; the long side is where the scale lands. The Erechtheion's porch of maidens sits on the northern lip, over the whole sprawl of the city.
Come down and finish on the Areopagus, the bare court rock west of the entrance — free, open, and the best place to sit and look back. Monastiraki and the metro are a ten-minute drop downhill.