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Fuhlsbüettel is 6 miles (10 km) from the center of Hamburg. There's useful information to be had on the airport's Web site, including:
The taxi stand is just outside the door of the arrivals area. The fare is metered and will cost about 25-30 deutsche mark ($13-$16). Travel time is about 20 minutes, and the taxis are comfortable Mercedes.
There is a very good bus service called the Airport-City Bus. From 6:30 A.M. to 10:30 P.M. it leaves every 20 minutes from just outside the arrivals area. The bus stops at the Hauptbahnhof (central rail station), and a few hotels. The fare is 8 deutsche mark ($4) and the travel time is about 30 minutes.
If you have very little luggage, there is a bus service that connects to the U-Bahn and S-Bahn which will take you to the city center. Fare is about 3.20 deutsche mark ($2).
Major companies are represented at the airport including Avis, Hertz, Europcar, and Sixt-AG.
World Airport Information Main Page
Adapted from How to Get Out of the Airport