Cheapest would be by coach (Eurolines via Copenhagen for instance) but that takes quite some time.
Lufthansa flies to cities in the north of Germany from where you can catch a train (Hamburg, Hannover) and Welcome Air is another option: http://www.welcomeair.com/
Then there is of course train but that's not very cheap, just nice.
Finally, you can catch the evening ferry to Kiel and travel from there to Bremen. Very convenient and relaxing but price depends on how many share a cabin, what cabin and so on. www.stenaline.com
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