A few suggestions:
First, check the Iberia website,
www.iberia.com When last I checked they were still running phenomenal deals on tickets (28 euros one way). Now, this will mean that you come in at the airport (a long way from the Plaza Mayor) and would probably need to take a taxi. But with what you're saving on the plane fare, maybe that's do-able (not to mention the time that you'll save!).
The bus station is probably a bit closer than Charmartin--I think you'll come in at Avenida de Americas. There, too, if you don't want to take the Metro, you'll need to take a cab.