Is there a station at or near the Barcelona airport?

Yes-but only at T2, the old terminal where the budget airlines come. And its aa long walk to it via an often overheated skybridge. Ev 30 mins a local cercanias train-free on pass, but its gated and you have to consult counter to oppen-if noone there-bad luck.

These train run via Sants=major station, where you can change to the hourly AVE trains to MAD-after having done again a luggage check and airline type securitiy. You should in theory be able to get the addit ticket for the MAD train at airport-but good luck for that

The airbridge BCN-MAD was once the very busiest in EUR-3-4 planes/hr so this trip looks rather ´not well planned!