I wouldn't really do the metro. Not only is it prone to disruptions, both planned and unplanned, it's also not on par with a private car in terms of comfort or privacy. It being Saturday evening, there's bound to be a good number of people on the Metro, people going out etc. and especially if there's a Caps or Wizards home game and you're on the wrong line at the wrong time it could even be packed. I guess it depends on whether an 'urban experience' is what the OP wants between two flights or if he wants something a bit more pacific.

Given he suggested a rental just to see monuments and landmarks, it sounded like it was more about picturesque impressions of the nation's capital. The metro is many things but it's not picturesque and it's unlikely to show the capital's best side.