Too bad you don’t have more time. I’m actually moving there once my visa gets processed.
Anyway, do the harbor bridge climb just before sunset. You get to see the city and the harbor in the daylight on the way up and then the city lights on the way down.
Bondi and Manly are cool.
Hike, bike up to the lighthouse at the entrance of the harbor.
Find the bar that you can play the bartender in rock-paper-scissors to win free drinks. I think it’s near the American University.