At first, you have to get cyrillic fonts, compatible with your OS.In Illustrator (Character palette) choose any of them and start typing.
Another thing is - how comfortable you are with standard keyboard layout.Without special efforts you will be able to get russian "f" under english "q" key and russian "v" under english "e".This is standard Russian keyboardind.If you used to type in Russia - that is for you.I prefer to use Phonetic keyboard - that means,english keys A,S,K etc.give me russian equuvivalents.
Andriy.