You are in for a treat!
I hope you like walking as it is the only way to take in the beauty of this great city. Check this site for 3 great walks.
Bondi to Coogee (via Bronte) walk is a must to get a sense of why there is such a strong beach culture in Australia. You should also try to get to either North Head or South Head to understand why the first colony was named New South Wales.
As for exhibitions the gallery scene is not great in Sydney. The only dedicated fine art "analog gallery" recently closed. There is a good exhibition currently running at the State Library featuring one of Australia's best known photographers, Max Dupain.

Some other galleries worth a visit but they are not downtown:
http://www.blender.com.au/ (does not always exhibit photography)
http://www.artgallery.nsw.gov.au/exhibitions/current (a surrealist themed exhibition running at the moment. this gallery is well situated and you can incorporate this into a walk from the opera house around the harbour and through the botanical gardens.)

I'd be happy to meet up with you during your visit if the work schedule is not too demanding. As it turns out I am heading to London on Sept 22 probably about the same time you are returning. Please feel free to PM.