Looking at the timetable at www.bahn.de it seems that all trains between Hamburg and Berlin are now ICEs.

If you type in "Hamburg Airport" as your point of departure you can see the whole itinerary including the bus to the station. (It says there is an 8 minute walk from the bus stop to the station, it's actually much less than that as long as you know where you're going)

Added (after reading last post) - second class is perfectly adequate, but with people travelling for the cup final you may find it's fully booked with standing room only. The 50% supplement for first class is probably worth it in your case, especially after a long-haul flight.