I must step in here and third the sentiments of JDR and taravb: the Metro is the ONLY way to go. And it really is a breeze. In your case, you'll get on Line 8 (Aeropuerto) and take it to Line 4, getting off at Mar de Cristal. Look for the signs that say Arguelles -- that's the direction you want. It's a pretty good jaunt from Mar de Cristal to Bilbao, where you'll get off and transfer to Line 1. At this point, you're home free. You'll want to hop on a car going in the direction of Congosto. From there it's just three stops to Sol. Remember, on the Metro, the last stop represents the direction of the line.
If you plan to use the Metro frequently in Madrid (and you should) buy a Bonobus (745 pesetas) instead of a single ticket (145 pesetas) as it gives you 10 rides.
My wife, two teenage daughters, and I traveled all over Madrid by Metro (including from the airport to Anton Martin, two stops past Sol)just a month ago. It's clean, fast, safe, and an amazing place to people watch.
Hasta luego.
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WOM