Your browser does not support (yet) fully functional text input/editing on a page that in a situation like Internet-cafe simulates Cyrillic text processing:
you may see that next symbol, generated by the script, can be placed only at the very end of the text and that the cursor is not moving during such operation, that is, it's not possible to edit/correct your text. Or you may see that not just a Russian letter appears on screen but also the English one which was pressed on keyboard, etc.

Thus if you'd like to work with text normally, i.e. to be able to insert, replace, delete at the any position in the text, have cursor working Ok, etc.
then you need to use - just for Cyrillic text input - a different browser - one of those that do support fully the imitation ("as at home") of input/editing of Russian text.

Please read about supported browsers and why it really does NOT matter that some browser (say Safari 2 or Opera 7) does not support normal input/editing -
on the Virtual Keyboard documentation page (opens in new window):

Usage info and comments