I know I'm a bit late in replying on this thread, but to be honest, I'd never even seen this whole section until now.
If you haven't already been by the time you read this, I would definitely suggest you do the Bridge climb. My wife had real fears about it, worrying about the height and the difficulty of the climb.
The height didn't bother her and the climb id done in stages, so you are never out of breath or physically tired.
The hardest part for people with a fear of height is in the early stage of the climb, when you cross over the road. The guide simply warns people with a fear not to look down. This worked for my wife and several other people who had admitted a fear of heights. We had no-one in our group back out of the climb.
As you climb the bridge section over the water, the climb is not steep and the guide stops at regular intervals to point out places of interest and give you some history of the area.
There are also several photo opportunities, although only the guides are alllowed cameras. You will not be allowed even a wrist watch on the climb, as all loose objects, including coins, are left behind in secure lockers.
You climb the bridge in a suit provided by the company, and you are attached by a safety line to a rail on the bridge at all times from outset to return.
If you do take up the challenge, I would suggest paying a little extra for the twilight climb. You set off in daylight and the sun sets as you climb, giving you a great night view of the city on your way back down. I really hope you get to try it.
As for other things to do, Chubby has covered most. I did get up to the Blue Mountains and stayed overnight at Bilpin Springs. Our weather wasn't great, but that didn't ruin the experience. You can go on a day trip, which includes visiting the Three Sisters.
The Rocks area is vibrant and a pleasant place to stroll around and find a sopt to stop for coffee, beer, lunch or whatever takes tour fancy.
We also went on a Captain Cook evening cruise around the harbour. I booked this in the UK before leaving and must admit to wondering how you could spend several hours cruising around a harbour. Only one I got there did I realise that Sydney Harbour has around 27 miles of coastline.
The cruise included seven course meal and all drinks. We both thouroughly enjoyed it and found it easy to mix with others on the cruise, particularly Japanese people for some reason. They were all very friendly.
As I said at the start of this post, I hope you have not been and returned before reading this. Please let us know if you do make the climb.
JJ