Currently there are no train services to/from the Hamburg airport to the town main railway station.

You would need to take a bus transfer. One runs every half an hour and they are called JASPER buses. Cost is €5 one way. The buses stop right in front of the main railway station. From there you can take a train (there are hourly ones) to Bremen.

Go to: www.bahn.de

Choose the english option (Upper right corner)

Go to :Time Table - Tickets

Start and Destination. Type in word HAMBURG HBF
And the TO: BREMEN

Type in date, and time....

Leave everything as is...

and Seach Connection

You will see options and train times. Prices will range between €20 for the ME which is the regional train, to €24 for the IC. The latter is faster and more direct as it won't stop at every station along the way. You can see that the IC trains run almost every hour at a quarter to the hour and takes 55 mins to Bremen. You may click on Maps/Route to see the route that the trains will take too.

Because the ME are regional trains, you do not need to make a reservation. You may just buy a ticket at the railway station and travel on that day. Same goes for the IC trains. Reservations cost about €2 more. If you are travelling at peak times ie rush hour, it will be good to get a seat reservation.

You may enter all information and book your ticket and print it our online. Yes you may use a cc that is not from Germany.

Hope this helps