Unfortunately Dulles is far outside of Washington, and so the hotels around the airport tend to cater to people flying in / out of the airport and business travelers visiting the many companies in the Dulles tech cooridor. I am not aware of any with tour desks that would run regular tours to destinations like Mount Vernon (about an hour's drive from Dulles on most days due to traffic) and Monticello (about 2.5 hours from Dulles). The only thing worth visiting that is close to Dulles is the Udvar-Hazy Smithsonian Air & Space Museum annex, which is on airport grounds and has on display things like a space shuttle, concorde, and SR-71 spy plane.

If I had to guess, your best chance for a hotel near the airport that meets your needs might be at either the Dulles Hyatt or the Westfields Marriott, which are two of the more upscale hotels near the airport. I'd call them directly to ask about whether what they offer in the way of tours will fit your needs.

A better plan may be to give up on the free shuttle idea, and pay for a taxi into (or near) the city when you arrive. There are many, many tourist-oriented hotels closer to Washington. Search on the Washington, D.C. forum and odds are you can find one to fit what you want.