The Aquarium is open 9:30-10pm daily. The entrance fee is expensive (US$14 for student discount!!, full price is probably closer to US$20) but there are discount coupons in every tourist pamphlet I saw in the tourist office so grab a pamphlet from the tourist center in Darling Harbour before you go. I know the price is high for a family but I think its well worth it. There is a lot to see and the kids I saw walking around were MEZMERIZED by the place. So was I!!
If you can't get to the Great Barrier Reef while in Australia, definitely go to the Sydney Aquarium. You will get an approximate experience that is uniquely cool unto itself. Beyond regular aquarium displays of small fish tanks full of marine life the Sydney Aquarium offers three giant underground tanks: one for seals, one for sharks and stingrays, and one that recreates the reef environment. You get to walk through the tanks and watch sharks, seals, and fish swim above you and around you.
If you are not traveling with kids, you can go at night when things are calmer. It's open until 10pm every night (last ticket sold at 9pm). The seals may be asleep, but all the other tanks are full of life at night and prove to be eerier after dark. I think the Aquarium is a MUST-SEE when in Sydney.