I guess it would sound the following:
"Отче наш, сущий на небесах! да святится имя Твое;
да приидет Царствие Твое; да будет воля Твоя и на земле, как на небе;
хлеб наш насущный дай нам на сей день;
и прости нам долги наши, как и мы прощаем должникам нашим;
и не введи нас в искушение, но избавь нас от лукавого."
(You can find this one in this great site:
http://www.christusrex.org/www1/pater/JPN-russian.html).
I would also like to point out that Russian Orthodox Church still uses Old Slavonic (
Церковнославянский язык) in liturgy, but Russian might be used in individual prayer, I guess. As for the other prayers, I would like you to suggest you to look up in Russian Wikipedia, because I do not know a better place to find them.