Sorry, no onboard access for the NASM's Concorde. (You can, however, tour the one on display at the Museum of Flight in Seattle; I highly recommend a visit.) I share your first and third regrets, FWIW.

There's a bus from the airport to the museum that runs about every 45 minutes to an hour; sorry for the overload of Flash at that site. Take either the Metro 5A bus to IAD from L'Enfant or Rosslyn ($6) or Metro to West Falls Church and then take the Washington Flyer shuttle from there to IAD ($10 each way, $18 roundtrip).