As the two airports are 30 miles apart I know of no hotel with a free shuttle serving both.

I would stay in Crystal City, across the parkway from DCA, on arrival -- a shuttle will come fetch you from the airport, though some stop running at 10p or 11p. Your flight is supposed to arrive around 719p though. Next morning, use Metro to get from Crystal City out to Dulles for cheap (Blue Line Crystal City to Rosslyn, change to Silver Line for Wiehle Avenue, then pick up the shuttle bus to IAD from there).

Your flight to NRT, UA7941, is actually operated by ANA and does not depart Dulles until 1240pm, giving you plenty of time to get out there in the morning.

You can almost always Priceline a 3* or 4* Crystal City hotel for less than $100, and many are within walking distance of the Crystal City Metro for a quick morning getaway.