You can take the Trolley from King St to the metro. DCA is only few stops from King St. Trolley is Free! Good luck.
No shuttle
You can take a metro from airport to King Street metro in Alexandria. Then Alexandria has a free bus that goes around the xity and it goes near the hotel. Check the route and hours at their site "King street trolley" operated by DASH
I don't know about a shuttle but it was incredibly easy to get an Ubet or Lyft ride at any time of day, even well after midnight.
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