You can't buy your ticket on the train. Your first train is a suburban train (S-Bahn), and not only is there no facility to buy tickets on board but you get fined if you are caught without a ticket. There is no barrier, and there are no regular inspections on board, but if you get caught in one of their occasional spot checks you're in trouble.

There is no ticket office at the airport station, and as hinted at in a previous post the machines aren't terribly user friendly. You definitely want to buy your ticket online if you can.