The tweet is based on the incorrect-but-common idea that "Duck and Cover" was useless for its time. It wasn't. For 1950, the fear was a very limited Soviet attack — usually with foreknowledge — of Nagasaki-style atomic bombs. Those would kill people within a relatively small zone of total destruction, but there were many more people in a zone of light destruction whose lives might be saved if they were away from windows, under desks, with their most vulnerable parts concealed. That is what Duck and Cover was meant to be — harm mitigation. It's not bad advice if you're in that particular kind of scenario. What changed is the underlying threat scenario: H-bombs, many of them, lack of warning, etc. Even with that, Duck and Cover still works if you're at a specific distance from the bomb, but the overall harm mitigation is quite reduced as the amount of harm goes WAY up for large-scale attacks. The guidance for Civil Defense, as this is called, changed in the 1960s away from "Duck and Cover" and to fallout shelters for this reason; they essentially abandoned the idea that cities would be savable, and instead focused on people who lived in suburban areas who would be threatened by long-range nuclear fallout from the new megaton-range weapons.

But anyway. To your direction question: yes, after Stalin, the Soviets did adopt Civil Defense programs. (Why after Stalin? Because under Stalin the Soviets were sort of officially kept from talking about nuclear weapons in a public way, in part because the official ideology was that they were not a big deal, itself a reflection of the fact that they were totally vulnerable to them and did not want to admit it.) In the USSR they also had versions of "Duck and Cover," but they also had far more extensive social infrastructure built up than was in the USA. Public blast and fallout shelters, for example, were pursued in the Soviet Union in ways that they were not in the USA (where their cost was deemed too high, and instead people were essentially told to build or organize their own private shelters). That isn't to say it that they would have "ridden out" a nuclear war any better — by their own estimates, their shelter capabilities were far inadequate compared to the (considerable) nuclear threat posed by the United States. But they went through the motions of it, anyway, as did practically all nuclear-armed nations during the Cold War. There is no simple or cheap way to mitigate the effects of a large nuclear attack, though

All of the "advice" given for Civil Defense was essentially similar, and based in both the facts of the weapons and what is known about mitigating their effects. (If anything, the US advice has had the largest influence, because the US did extensive nuclear testing to develop its policies and then publicized them in an unclassified form, influencing lots of nuclear powers.) "Duck and Cover" approaches are useful if you are in an area of moderate blast and moderate thermal. Fallout shelters are useful if you are downwind of the fallout plume. City evacuation in theory would be useful if you could do it with sufficient advance notice — always the tricky part. And there are a whole category of things to do after the bombs have gone off, to try and mitigate further damage, keep continuity of government, protect the food supply, etc. The basic idea of Civil Defense is that the bombs going off will not be the final "curtain call" on your society, that a significant proportion (50% or more) of your society will likely survive no matter what (not an unreasonable assumption), and that the degree of terribleness that comes next will be partially related to how good your planning was before it happened. So Civil Defense planners tried to anticipate what would occur and how you might mitigate it, to varying degrees of plausibility considering the changing nature of the nuclear threat over this period. We tend to mock this today, but it was not only the origins of modern disaster preparedness as we consider it today (the present Federal Emergency Management Agency was grown out of the institutional ashes of the Federal Civil Defense Administration), but it is arguably a pretty standard function of the state to try and anticipate these kinds of threats and prepare for them. That these states were also complicit in these threats was known to everyone then, but then as now there did not seem to be easy ways out of that particular dialemma.

Anyway, for more on the history of both US and Soviet Civil Defense programs, the most useful book is Edward M. Geist, Armageddon Insurance: Civil Defense in the United States and Soviet Union, 1945–1991 (University of North Carolina Press, 2019).