Hello. I studied Russian at the preparatory faculty for foreign students of a university in Kiev for an academic year (2011-2012, KNEU, kneu.edu.ua, go with a friend because only minimal English is spoken at the dpt. for foreign students). There are many universities which propose equivalent programs throughout Ukraine, in Ukrainian or Russian.
My estimated costs for the university year 2011-2012 were: USD 1400 for university fee (some other universities were asking for 2000-2200 USD, all of which will be higher now in 2013, avoid Schevchenko University which is at least twice as expensive for the same program) + about 1000 UAH for medical insurance (seems compulsory unless you can show prof of insurance) + cost of medical check-ups (2000+ UAH if you go to a private clinic where they speak English and much cheaper if you have someone speaking Uk/Ru with you and you go to a local hospital) + cost of compulsory HIV test + cost of translation and notarizing your passport and your previous degrees + cost of Visa in your country + all the costs related to the registration in Ukraine (you got only a 1 entry student visa and had to convert it to a residence permit and multiple entry visa it by registering with OVIR ; a new system of residence permits has been implemented since - they will know at university. Do note that it seems more and more difficult to avoid the university dorms because registrating at an apartment you'd rent outside of uni. seems increasingly difficult. Pl. note that the dorms at Schevchenko University for foreigners were also substantially more expensive than at other universities). Living expenses and traveling costs are not included here of course. I think I haven't forgotten anything substantial.
At the university of Trade and Economics, a French guy who was studying in 2010-2011 was telling me how the police sometimes came to check attendance registries at the university or the dorm (I can't remember which one). I don't know how strict they are for students who wouldn't come to class but on paper at least, your stay is conditioned by you going to class. At my university, teachers were good, it was quite intensive (3 hours / day, 5 days / week + homework) during the first semester... Second semester, we had also mathematics, geography, economics and computer science in Russian. Maybe you can skip them if you're only interested in the language, but they were other occasions to learn the language... in a different way. These other subjects were focused on giving students the basics tools and vocabulary to cope with the first year of university the following year.
And finally at any university they will tell you whether there's a law passed in 2012 which would prohibit foreigners born before 1986 to apply at university (probably in a bid to avoid many living in Ukraine as students but actually working). I haven't had confirmation of that law (only 1 source so far).
For short term study programs (3 months or less), I met many foreigners who were studying at Nova Mova, http://www.learnrussiankiev.com/
, check their prices on the website.
Hope it helps.