It is probably too late now, but American Airlines is part of the Rail&Fly program.
https://int.bahn.de/en/booking-information/rail-and-fly

Through AA or your travel agent, you could buy a ticket that includes both your flight and a train ticket to anywhere on the DB network (which includes gorgeous Salzburg in Austria). You could board almost any DB train the day or your arrival or the day after.

Rail&Fly also works in return; you could take a train from anywhere on the DB network to your German departure airport on the day of your flight or the day before.

If I am not mistaken, the IATA airport code for Rail&Fly is QYG.