Although this is a little late, I'll respond anyway. I've taken Grand Express (third class, which they call first class, but it was still rather nice and I got a much better night's sleep than I did on my transatlantic flight on BA in WT+), but I can't help too much with the booking, unfortunately. My ticket was booked by my university. I'm not sure if it was done by the US institution or the Russian institution (which was a division of the Ministry of Foreign Affairs) or even the program director. You may have to book through a travel agency because their Website says they don't sell tickets outside Russia. However, if you go to the Russian side of their Website, they'll let you buy a ticket online. They ask for a passport number and accept a "zagranichniy passport" which is one good for exiting Russia, but they don't seem particular on the country. You can tell them you're going to pick up your ticket it seems ("purchase ticket without delivery"). I'm not sure I trust this, though, because it's insecure and there's no actual method of payment, so it seems you still have to find some way to pay for it. I can't figure out how that works. Maybe you pay the messenger that delivers it...
Otherwise, like so many things in Russia, you'll have to go through an extra layer of bureaucracy, and in this case, it's the travel agent.