Can I stay in Russia and get a job after completing my studies?

This would not be easy. A foreign national’s legal stay in in Russia is secured by their student visa valid throughout the period of study. After graduating from the Russian university, foreigners cannot stay in Russia or land jobs if they have not obtained a work visa/permit. You need a work visa or work permit to stay and land a job in Russia. The list of required documents for legal stay in Russia after completing university studies is available on the website the General Administration for Migration Issues of the Interior Ministry of Russia. Also, you can work as you study at the university. More information.