Most universities have an on-site clinic (poliklinika) on campus. Those who do not will have a partnership with a nearby hospital or clinic.
Russian insurance will cover most of your expenses in policlinics. However, as a foreigner, you can get better treatment in private clinics. Private medical centers are available in the larger cities and towns and provide a range of medical services. However most of them require payment. Some take credit cards while others cash up front.
Remember there are no private doctors in Russia and you must make an appointment. A guide to physicians, clinics, hospitals and emergrncy services can be found on most consular websites.
Outside the major cities (Moscow, St Petersburg, and Vladivostok) the healthcare infrastructure throughout the country is fairly limited in terms of services offered and is only typically able to provide emergency stabilizing care.
Medical care in Russia is still below Western standards; shortages of medical supplies, differing practice standards and the lack of comprehensive primary care all combine to make the medical system difficult to negotiate as well as suspect. The few facilities in Moscow and St. Petersburg that approach acceptable standards do not necessarily accept all cases (i.e., they may not be licensed to treat trauma, infectious disease or maternity cases).