Under Alexander II, Russia underwent an era of reform - called the "Great Reforms". The main features of this were the 1861 Emancipation of the Serfs and the introduction of the zemstva and city dumas as a form of participative local government in both rural and urban municipal settings. Before these reforms, poor relief was provided by philanthropic efforts of the aristocracy and imperial family, a form of relief that continued alongside these reforms as a way of providing it that was palatable to these elite strata of society. The development of almshouses and hospitals devoted to workers particularly likely to encounter illnesses began in earnest around the time of the 1812 war, surrounding sentiments of nationalism and concern for soldiers returning home wounded, but secular charity dates back to the time of Catherine the Great, whose earnest celebration of Enlightenment ideals instigated the creation of orphanages, boarding schools, almshouses, insane asylums, and so on.

I am afraid I am short of specific information on hospitals, but the period of the Great Reforms saw Alexander II taking seriously the problem of Russia's great size and disparity of wealth within its population. The zemstva dealt with very seriously and with measurable effect the implementation of medical care, public welfare, and issues like road maintenance. With an enormous rural peasantry spread over great distances, and a handful of major cities dealing with slow industrialisation, the reforms arguably sought to placate rising tensions and unhappiness from "risk" areas of society, which was expressed in acts of terrorism. Ultimately this would include the assassination (by bombing) of Alexander II - this was not the first assassination attempt on his life by elements avowedly upset with the hierarchical social order of Tsarist Russia.

In response, Alexander III's social programme was much more conservative, and reversed many social reforms of his father, using his father's assassination even as justification. It is worth noting that whatever the prevailing sentiment behind governance, whether socially liberal or conservative, the governance itself was always authoritarian, and certainly existed to preserve the social order. Alexander II removed some more visible privileges of the aristocracy, for example by instituting universal military service, but the goals were to remove objects of resentment and remedy practical causes for unhappiness, rather than to genuinely destratify society.

As a thought about the dichotomy between "socialised" healthcare and "private" healthcare (the latter you don't mention, but I want to illustrate a point, since if the government does not pay for something, then a private citizen must), it's important not to think of this in terms of the modern American healthcare system, for example. One major thing Tsarist and Bolshevik Russia had in common was state management of the economy, and it's important to see state-run healthcare in this sense and with the wider picture of a country whose economy was steered by its aristocracy, not by what Marxists would call its bourgeosie - its middle classes. Unlike the great Western powers, Russia did not have a significant capitalist class, and the only way to implement things like healthcare was by state directive or by charitable acts of aristocrats - often effectively the same thing, merely implemented in a different way.

Hope this offers some perspective in rephrasing your question.