More specifically, I'm wondering about problems such as addiction and punishment or rehabilitation, the black market or underground economy, and general corruption. Basically, the problems that the USA is facing now that opiates are becoming so common/popular.
Can you be more specific in what you're asking? Which problems in Russia?
A lot of drug trafficking history is shrouded in conspiracy, mystery, and misinformation, because of the complicity of governments, NGO's and public figures worldwide. Additionally, it's difficult to get a handle on how common addiction was in a place like the USSR because of the historical unwillingness of the state to admit to the presence of a public health problem with narcotics - this UN briefing, for instance, claims that there wasn't a single case of heroin addiction in the USSR from 1961-1971 - a country that, by the end of the 60's, had a population of nearly 250,000,000. Considering the global rise in demand for opiates during that decade, this assertion is a little hard to swallow, especially with the USSR sharing borders with the opium-producing Afghanistan and Turkey, and heroin-producing Poland.