Cheap is not a word I would use for anything in Moscow. It is a beautiful city but the most expensive in the world.
I stayed in the hotel Ukraina while in Moscow. It was not cheap but well worth it.
Sorry Polo, I have been to moscow too, the apartment cost killed me, and it was the cheapest I could find(and small). All I can tell you is I got the apartment from the same office that verifies your visa when you are there. I do not have the link, sorry. But if you get the info on where to do the papers in moscow when you get there, they have about 10 apartments.