There are multiple language schools in any town, also any university that has international students usually has courses of Russian for foreigners.
Sorry, I did not have time to do an extensive search, but a quick search gave me this (prices from December 2010):

Odessa language school “Yeni Dunya”

For foreigners: a course of studying Russian and Ukrainian languages. Knowledge is presented with the qualified experts who have knowledge of English, French, German, Italian, Spanish, Turkish, Persian, Arabian and Chinese languages, which allows you to study the language quickly.

Prices
Small groups (up to 5 students) – 1 lesson (1.5 hours) – 75 UAH
Intensive course lessons suggest studying three times a week at the most appropriate time for you.
Private lessons - 100-150 UAH.

Special offer for corporate clients!
You can study Russian in the comfort of your office, as well as in our language centre “Yeni Dunya”-“New World”.
Our clients are: "Turkish Airlines", "Mugla Odessa", "Astra Grain" and international holding "Arkas".
I'm sure there are many more options.