The Doubletree is at Newcastle airport, which is 10 minutes from the city centre on the A696. Turn right at the roundabout into the airport complex (or left if you're coming along the A696 from the north), then take the second exit at the next roundabout and follow the one-way system round until you come to the hotel.
The hotel is also a minute's walk from the Newcastle airport Metro station or you can catch the 131 bus , which stops a two-minute walk from the hotel.