Trip to the city from Darwin airport:
When you exit the airport you'll see a reservations office with a shuttle service with direct service to hotels. Indicate the hotel you're going to and they will give you the tickets. The cost is 15 AU$ per person and takes around 20min. If you are a group of 4, take a taxi (25-30 AU$, 15min). You can also take the public bus n3, but it takes 60min to the city center.