Your plan looks great (if a little rushed).

I'd skip Cairns City - there's really nothing to see there. You are much better off spending the day at the rainforest. How will you get to Port Douglas from Cairns ? You'll need a rental. You can use the car to go the rainforest as well on your second day.

In Sydney US$250 a day is a good budget (when not paying for accommodation). I think in only two days I wouldn't bother leaving the city - though if you really wanted to stretch it you could do a blue mountains tour on one of the days.

Maybe do a lunch or dinner cruise (maybe start keeping an eye on Scoopon, Stardeals and Jumponit for Sydney as they often have heavily discounted vouchers for harbour cruises). Head to one of the beaches (Manly or Bondi) and take in the sites around the harbour.