For about 10 euros, you can take the public transportation - you only need to change once and the stop is right outside the door of the Welt. There's a special airporter fare that is good for unlimited travel for that day from the airport. Now, finding it on the machine is a little tricky.
Don't remember how long it took, but you can check the schedules. By far, it is the least expensive way to get there. Starting at the airport, you should find a seat easily. You're cutting it a little close, as one never knows how long customs/immigration and luggage will take, let alone any flight delays.
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