I realize that the overlapping period of time between the fall of the Soviet Union and the rise of the Internet as a social phenomenon in the mid-90s is small to nonexistent, but did the Soviet government at least experiment with global computer networking?
Their was a famous 'kremvax' hoax in 1984, but apparently their was a real VAX system in moscow in 1991 https://stuff.mit.edu/afs/sipb/user/marthag/text/humor/april_fools/kremvax
There were a few websites, I think mostly relating to universities. They used the .su domain name, which has a lot more web pages registered to it than you would think. Apparently Russia kept using .su for a little bit after the collapse of the Soviet Union, before they switched to .ru in 1994.
I don't really have a source for this besides wikipedia, though. You could probably just visit some of these sites yourself!
In 1990 they were assigned the .su country code top-level domain but dissolved fourteen months later. Up until that point most internet activity was done by the military and universities. One of the earliest domestic software networking companies still active in Russia, Demos, was founded in 1989. If you'd like to know more, someone asked a similar question a while ago.