Just wanted to update, since I did some research. If you want to take the metro, apparently, there is a metro station at the Reagan airport. You could take either yellow or blue line from there; both would go to Alexandria. Dulles wouldn’t really work, since their shuttle goes to the silver line and you cant transfer to the blue line until Rosslyn. It’d be a much longer metro ride.
EDIT: Crossed comments with you. I have several friends in NY at UR. You could actually take Amtrak all the way down from NY and avoid the flight all together; that’s what my friends do- its actually faster. I would look into seeing whether there is an Amtrak station near you or whether you could get to one easily and explore that route.