How to get to our Sydney hotel

Perhaps you could email the Sydney Tourist Office for this info.

Thanks - I believe I found out how to do it via Goggle maps !

there is an airport train that gets you into the city within short time and very easy, as the airport is right in the city. just follow the signs inside the arrivals building, very easy.
if there are a number of you travelling together it 'might' be marginally cheaper to catch a taxi? ( need to check) as the airport train component of the trip ( into central station) is more expensive than the usual city section of the journey. if you are travelling by public transport, Sydney and NSW uses a Opal card , which works well. ( like those used in other major cities)
http://www.airportlink.com.au/opal-cards/

given that you will be near darling harbour, and need to walk from the nearest train station, perhaps just as easy to catch a taxi.
wynyard station 'might' be the closest? the walk would be generally downhill, if that helps.
http://www.airportlink.com.au/maps/wynyard-precinct-brochure/

hope this helps.

Depending on your hotel,
Wynyard station is the most likely station closest.
Short walk, Up Margaret street, or Erskine street again depending on the hotel
In October, the Margaret street route had a bit of a detour, due to construction of a mall under the
Expressway, was well sign posted and even the escalator down was in service .
Regards

The Rocks is a nice area to walk around. The hotel I stayed at had a rooftop with beautiful night view of the bridge and Sydney Opera House. I took the train back to the airport, catching it at the Quay. I purchased a card similar to the Oyster Card- easy transport. Enjoy your time in Australia!

If you go with the train it is just under $20 each as the airport is a private section. You would go to Wynyard station then less than 1 Km walk.
With more than 1 if you it is most likely cheaper to take taxi or uber at $30-45.

Good advice from glennlorrainer that, in this particular case, you're probably better off just springing for a taxi to get you directly to your hotel. The minimal cost saved by using the train to Circular Quay isn't worth the hassle. There's a lot to be said for the comfort and convenience that door to door service can provide.

Not to digress from your original question, but one pleasant discovery we made on one of our first trips was the availability of showers in the T1 (International Arrivals) terminal. It was always refreshing and rejuvenating to be able to shower, shave, and generally feel human again before striking out on our first day ... especially given the early morning arrival time after a long (long) transpacific flight.
The T1 Arrivals Map can be seen here if you're interested: http://www.sydneyairport.com.au/find/airport-maps.aspx

Thanks everyone for your useful information !

I used the airport shuttles all over Australia, including Sydney.
I don't know why no one else here has mentioned airport shuttles, so I am feeling a little timid making the suggestion to use one. Therefore, I guess I will just suggest looking into it. I just now Googled "Sydney airport shuttle" and did see the one I myself used. Then, the hotel I stayed at turned out to have their own shuttle, a little cheaper.
Anyway. It's another option.

After a 15 hour all night flight an airport shuttle for 15 AUS Dollars is money well spent. They can be arranged for in advance or just go outside and walk 200 or 300 yards to the right after you exit security to the shuttle bus area and wait for a shuttle. I was just there this past month and that is what I did. There was a guy in a vest that pointed you to a driver of the next shuttle bus to leave for town. Or search for one on the internet in advance.$15 AUS seemed to be the going rate. Dragging your bags onto the train and then walking to you hotel while jet lagged is no way to start a visit to anywhere. I got a return shuttle ride back to the airport for $14 AUS which is about $10 US. Money well spent IMO.

Taxi will be approx $50.
Train fare is $18.50 ea. (premium fare as it's a private line)
All things considered, with 2 people, I would definitely jump a cab.