Rudi,
The Airport shuttle from Canberra Airport to the city will not take you to the Hyatt - the closest it comes is 2.5 kilometers away.
The Shuttle goes from the airport to the commercial centre of town on the north side of Lake Burley Griffin.
Canberra is divided North/South by Lake Burley Griffin - the North side is basically commercial, while the South side is government. The Hyatt is near Parliament House on the South side.
The shuttle takes you to the north side of Lake Burley Griffin, and, if your hotel is on the north side (eg: Novotel/James Court/Saville/Crown Plaza) the driver will do a drop off to the door of your hotel if he's not too busy.
The Shuttle costs AUS$5 (US$2.50) each way and is great value if it's going where you want to go.
I am going away in a few days to spend some time at Uluru but should be back by the 30th and if so would be delighted to pick you up from the airport, though sadly I may not be able to join you for a night-cap.