@Richmond Garner is right that you can take the subway from Estación Sur to Barajas, but you will need to change. The metro map is kind of confusing, but the system itself is amazing.
Magnified view of the Madrid metro map system. View all 12 lines and all stations along those lines.
www.madrid-tourist-guide.com
As rome2rio suggests, the best subway route would be to take the Circular to Nuevos Ministerios and there hop on Line 8 direct to the airport.
You have to know whether your plane leaves from T4 (Iberia and One World Alliance partners like American and British Airways) or T1, 2, or 3 for all the rest.
Good luck! Public transport in Madrid is awesome, so you should be fine. Buen camino, Laurie