As a followup to my previous reply, and also your post about St. Petersburg, you might do a Google search for a private car from downtown Moscow to the North River Terminal.
Also regarding use of the subway, I don't recall the station closest to the North River Terminal. It is a transportation center with buses and I presume taxis. The walk from it to the North River Terminal involves going through two park areas on either side of a rather busy street. To cross the street, there is an understreet walkway with steps that you might find difficult with luggage. Once you reach the rather long dock area, not knowing exactly where the Tikhi Don is located could mean traversing some distance with luggage.
Again, I hope this helps. Russia Revealed is a fabulous tour.