We flew in the day before boarding. Unfortunately, it was raining and we didn't do as much as we'd hoped. We did walk around the area of the hotel, found a food court for lunch in a mall created in an historic building.
Later we took a local bus to the Marriot for an early get-together of people on our CC (the other cruising website) roll call. We ended up walking back and finding a pub for dinner.
After lunch on the ship the next day we went across the street to The Rocks, an open air marketplace and did some browsing and shopping with another couple we met last year, also on Mercury. The famous opera house was right across from our cabin window. We opted to skip the bridge climb, but if you want to do that you should get reservations in advance. The Hilton had an X rep on site and the concierge was very helpful.
There is a Hop On/Hop Off sightseeing bus you could take since you have more time than we did. Our cruising friends came in the day before we did and had booked a private tour to the Blue Mts. and Featherdale Wildlife Park, instead of doing the cruiseline tour on sailing day like we did.
Check this and the other cruising website to see if there is a roll call for your cruise. You may be able to hook up with someone and roam around together.
Don't forget to get your Australian visa before you go. You can do it your- self from the visa (ETA) website for $20 AUD or your TA can do it for you at no cost. There is a $25 pp departure tax that must be paid before you leave NZ. We used NZ cash, but you can use a credit card. They seem to get you coming and going, lol.