St. Petersburg State University was founded in 1819, but but claims to be the successor of the university established along with the Academic Gymnasium and the Saint Petersburg Academy of Sciences on January 24, 1724 by a decree of Peter the Great. Peter the Great imported many "Western" ideals of time to what was a pretty insulated country.

Under Peter’s system, secondary education took place in gymnasia, which were later divided into boys and girls schools. Students attended the gymnasium for 10 to 12 years. Initially, only those in high social circles could afford to send their children to these schools. Later, in the late 18th, early 19th centuries, schools began to open to the general public.

Starting from the middle of the 18th century, higher education as a system started with the foundation of universities in Moscow and St. Petersburg. The system was modeled after that of the Germans: it was open to secondary school graduates and took five years to complete. Upon completion, a "Diploma of Higher Education" in a specialized area was granted. Since then, the system itself has not changed much.

So before this time, and linked to your question of midieval Russia, kids of wealthy families would be taught by a tutor, probably imported from a more educated country to teach maths, sciences, etc. Poorer kids, if they could spare the time may have gone to a local school, but this was probably pretty rare - people were pretty busy and kept their kids busy when in came to farming.