We traveled on the Russia Revealed tour two years ago. Knowing the area around the River North Terminal, I suggest staying at a hotel near your airport rather than near the terminal. Airport hotels will be more used to dealing with international travelers, more likely to be conversant in English, probably have a shuttle to and from the airport, and could arrange a private car the following day to the terminal. Use TripAdvisor to select a hotel near your airport, searching on Moscow and the airport name.
If you do this, insure that you select a hotel near your arrival airport. There are two airports that serve Moscow: Sheremetyevo, which serves international flights, and, Domodedovo, which I understand serves flight from within the continent. We flew direct from Atlanta to Sheremetyevo, but many on our tour who changed planes in Paris, Amsterdam, etc. flew into Domodedovo. Make certain that you know which is your arrival airport.
Another possibility is the purchase of a transfer from GCT, now that they offer them on selected tours. Stay at the airport hotel and then return to the airport the next day to meet the GCT rep there to greet arriving tour members.