Taxi fare from the airport to downtown is about 30 euros, and about the same amount to return. As you have plenty of time and don't have to deal with large luggage, the metro sounds like a good choice. I've never tried it, but if you do please post how it works out for you.

The museums are very interesting, but I would be trying to stay outside to help with the jetlag. There is a very nice large park (Parque de el Retiro) behind the Prado museum. Also a decent park between the Palacio Real and the river. Eat lunch in Plaza Mayor and stroll around that area.