Theres a school in Moscow called ЦМО. Its actually a branch of MGU that specializes in teaching foreigners Russian.
I studied there last year and I'd highly recomend it.
Center For International Education :: Lomonosov Moscow State University
That is their website. They'll give you everything from the invitation to a place to live. Though if you want to live on your own you can do that too. I lived in my own apartment.
The cost is about 1,000 dollars for 3 months, 3 times a week, roughly 8 hours a day. You can study 5 times a week as well but its a little more expensive. Besides, you might want more free time to see the city, make friends, etc.
Anyways its a good stepping stone for opening up Russia to yourself. I was recomended that place by a friend and now I'm recomending it to you
Good luck
Also, there is no due dates for joining and no applications.
You can join any class at any time, all you will do is take a placement test when you get there so they know your level. But don't worry about showing up in february and thinking that you'll be behind or whatever, many people come and go regularly. So yeah, there ya go.