I hear generally good reports about "snore and roar", the overnight experience at Taronga zoo. Given the high price, we havent done this ourselves. Living in Sydney, it seems a bit pointless.
As for construction work, there is always some minor repair happening here and there, but no major works at the moment.
I really dislike the "zoo" at Darling Harbour. Its depressing. (The aquarium is pretty good though).
Given that you will probably have a fair amount of luggage, the easiest way is definitely a taxi from the airport to the zoo. (Confirm when your checkin time for the hotel would be. But you can drop the bags and explore the zoo).
Depending where your ship will depart from, there are different options. White bay: taxi.
Circular Quay (overseas cruise terminal): Ferry from Taronga wharf to pretty much the boarding area.
Bus 100 from Taronga zoo main gate to Wynyard station (the last stop) and a 5- minute taxi ride from there or 15minutes level walk.
(Taxi or Uber wherever i say 'taxi')