I don't know if they offer a shuttle to the airport, but I saw this on travelzoo's site and the solmelia site says that the hotel is located next to the airport:
Solmelia.com is now offering a special rate of 50 EUR at Tryp Diana 4* (Madrid). This offer is valid for travel now through May 31. Exchange rates can fluctuate, but that's about $65* USD
Price per double room per night: from € 50 + VAT on a room only basis. Only available through solmelia.com
To Book: Click
here for more information on the hotel and click "Book Room" to reserve online. Or, call 1-888-95-MELIA (1-888-956-3542).
-Linda