The more I looked into this, the more I became certain that there is an entire book to be written just on the history of alcohol and alcohol regulations in Russia — and there actually has been! Vodka Politics, by Mark Lawrence Schrad. However, Schrad is a political scientist, and even if it were a history, it would be inadmissible as an answer on its own because of the rule against presenting sources or quotations without analysis, so I'll just throw the link out there and in the meantime get on with my answer.
The thing is, unless I've overlooked something, anti-alcoholism measures as such did not exist at all in official capacity in 1860s Russia. There were prohibitions on certain kinds of alcohol around that time, but the reasoning was not based on any impulse toward temperance, and consumption was generally encouraged. There was a temperance movement in the country at the time, but it didn't have very much traction.
Alcoholism in Russia in Dostoyevsky's time was not statistically as severe as it is now, for sure, since the per capita consumption was lower, but alcohol's importance to the average Russian, be they a serf or a city laborer, was not much less than today — it was just as central to daily life, just weaker and better moderated. Russians had traditionally drunk weaker forms of alcohol than vodka, but by the 1860s vodka had pretty decisively become the main drink on the market. The key was taxation. Alcohol taxes had always provided the tsar with a relatively stable source of revenue, and its cultural importance grew over the centuries preceding, so it would be wrong to say that any one ruler decisively introduced or forced vodka and alcoholism on the Russian people. It was under the reign of Nicholas I, though, in the 1840s, that the financial pressures of expansion into the Caucasus, industrialization, and the construction of the Saint Petersburg–Moscow railway became significant enough that beer production was ordered to cease and beer sales prohibited outside of Moscow and Saint Petersburg, with the aim of encouraging sales of taxable vodka. Consumption of vodka, mostly government-produced, is widespread at this point, but drunkenness would hardly have been a crisis.
Edit: I feel I understated the case a little here. Drunkenness was not yet an epidemic, but extreme binge drinking certainly did occur in noticeable numbers, and vodka was already being described by outsiders as one of the state's methods of maintaining a subservient serf population.
One reason that vodka consumption remained in relative check during the 1840s and 1850s was the very reason its production had been emphasized: its taxability. Vodka taxes provided an excellent income, but the system of taxation remained archaic, based on a tax-farming system that serfs disliked quite strongly, and which threatened the stability of the arrangement. When Nicholas' son Alexander II disturbed the balance in the late 1850s by raising taxes on vodka further, serfs protested by refusing to drink, and in some cases even attacked taverns. This "Sobriety Movement" of 1858 and 1859 was therefore, actually, not at all related to the temperance movement.
At any rate, the sobriety protests forced Alexander to reform the system of alcohol taxation, replacing tax-farming with an excise tax in 1863. This reform also included allowing many more private plants to produce vodka, taking the monopoly out of the state's hands. Another thing to note is that serfdom was officially abolished in 1861 (it could be argued because of the sobriety protests), which technically introduced much greater mobility into the economy. Practically, though, it did little to give freed serfs independence of their old masters, in many cases leaving them in immense debt. So vodka suddenly became much more available, taxes on it were rationalized and reduced, serfs had a much greater incentive to drink, and city laborers were suddenly under more pressure because of the loss of job stability. Drunkenness suddenly became a mass occurrence. I can't say for sure if taverns "overflowing" with drunks was common, but the increase was concerning.
This is probably a strange parallel to note, but I'm almost reminded of the Chernobyl accident 125 years later — a poorly-informed attempt to deflate a moderately worrying problem suddenly explodes into a terrible public health crisis much, much worse than the original issue. The rest of the parallel is not at all accurate, of course, but I had fun thinking of it.
It's also important to mention that this sudden epidemic of drunkenness was not at all long-lasting, or even really quite as terrible as it probably seemed at the time. Schrad makes the point that average per capita consumption of alcohol actually decreased between 1863 and 1914, when Nicholas II prohibited all alcohol as part of the mobilization for WW1. So the explosion of intoxication in the early-mid 1860s was offensive enough to reactionary moral types like Dostoevsky — his original plan for the novel that became Crime and Punishment was titled The Drunkards — but in the long run, it cooled off pretty quickly, largely because the novelty of the price drop and increased supply wore off. It also didn't hurt that the price of a liquor license increased by over 100 times over the period between 1865 and 1885. 1885 was also the year that village assemblies were officially empowered to close any alcoholic establishment they chose, and they seem to have used that power, if not excessively. So the problem which had seemed so horrible in the 1860s was solved pretty effectively within a generation, partly through better policy-making and partly just on its own.
I'll also just take a brief digression to talk about that temperance movement. As I mentioned, the movement had very little actual role in any kind of alcohol regulation or taxation at the time, but in terms of rhetoric the movement was pretty similar to its incarnations in other European countries and in the United States. 19th-century Russian medical conceptions of alcoholism, like in the west, viewed it as a curable disease, although, also like in the west, less scientific rhetoric did often paint it as a personal failure of character. And yet again like in the United States and Europe, religion played a role in the movement; the Orthodox church was responsible for organizing some of the first societies for sobriety. However, its hand was pretty limited by the fact that it, like the state, benefited pretty well from the vodka monopoly and the tax-farming system, and its role was probably more of a hindrance than an actual supportive one. I don't mean to just list points of similarity, because you might have assumed all this, but I found it interesting to note just how much Russia and its European counterparts had in common in this regard.
Sources:
Морогин В.Г., Костина Н.П. Социально-психологическая история алкоголизации России // Медицинская психология в России. – 2013. – T. 5, № 5. – C. 14. doi: 10.24411/2219-8245-2013-15140. (Accessed here. Yes, it's in Russian, but there's no English translation of it yet, so there's kind of no point to giving a translated citation, I feel. It has some narrative flourishes of Slavicism that I don't like, but it was a useful source.)
Schrad, Mark Lawrence. Vodka Politics: Alcohol, Autocracy, and the History of the Russian State. Oxford University Press: Oxford, 2014. pp. 117–124. (A popular history, and Schrad is a political scientist, but I still found it worthwhile, especially considering how much the article above seemed to be pushing a narrative.)
Mochulskij, Konstantin V. Dostoevsky: His Life and Work. (I used the Russian version available online, but the English translation is by Michael Minahan, Princeton University Press: Princeton, 1967.)
"The Sobriety Movement". In: The Great Soviet Encyclopedia. (Accessed here.)
In case you're interested, here's an answer that takes the anthropological and ethnographic approach to the history of alcohol in Russia, which is quite good, and which I used as a source as well.