G'day ChicagoD, welcome to FlyerTalk and the Australia/NZ forum.

Your desire to fit three activities in say five hours is ambitious but may depend on the timing of the SOH tour. My mother and sister arranged a tour a while back; the one they recommended begins quite early IIRC, so check if you can reasonably get to Circular Quay in time for that early morning tour.
I believe they took the Backstage Tour.
There is also the Essential Tourand possibly others.


Sydney is almost as good an architectural tour destination as Chicago.
I believe there is a fair amount of work still happening on interior renovations, closer to the architect's plan, in progress at present.

As you may have noticed, we get these kind of questions here all the time, so search is your friend.

If the weather is good, go to the Royal Botanical Gardens, adjoining the Opera House.

Between the other side of circular quay and the Harbour Bridge pylon, it's worth walking through the historic streets and alleys of The Rocks. I recommend you have a drink at Blu Horizon at the Shangri La, about 40 stories up and with a great view of the Bridge and part of the harbour. What a view, what a nice place to sit and watch. If you can't get into the Pylon, this is an even better view IMO.

There are some good places to eat around Circular Quay, although almost all are, naturally, touristy. Put Customs House on your list, even though it is relatively expensive, for its amazing view.

Please search for earlier, similar questions in case you find some other options that you might enjoy. Many people catch a ferry to Manly if there is time, or go somewhere near the areas I've just mentioned.

There are plenty of options, so first see if you can be at the Sydney Opera House at the right time for a tour and work around that. I'm sure you will get some answers here, but you will find a lot of good (general activity/recommendation) answers already here in previous threads.