Cowboy1968

The fully-flexible fare for a single ride from Frankfurt Airport to Aachen in 2nd class costs €83.
You can buy this in advance - print it out or load it on the app/ your mobile, or walk-up same day to a machine/kiosk at the train station - it will always cost €83.
If you buy it at the desk, it may cost a few € more for dealing with a human being.
The smartest way is to buy in advance so you won't have to deal with kiosks or queues after a red-eye flight, and just hop on the next train.

As your brother will probably know when he needs to return from Aachen to Frankfurt, you can buy THIS single ticket in advance to make use of the discounts.

PalenQ

Or book a discounted ticket from airport but build in a lot of fudge factor time -Frankfurt Airport has many shops and restaurants to help you recover after all-night flight. These can be as low as 29 euros I think so if your plane is really late you are only out 29-39 euros and of course still have to buy them a full-fare ticket.

Pretty safe bet if you put 3 hours or so extra.

In any case having a ticket in hand helps hurry the process a bit.