I tried looking for Russian Science Fiction from the cold war but can't seem to find any? Is there actually such thing? I just wanted to see examples of Communist Science Fiction or how their science fiction is like.
Although the original question focuses on the Cold War, please allow me to extend the scope of the issue to the entire existence of the USSR, as there are some interesting Soviet movies that predate the Second World War.
Arguably the first science-fiction movie made in the newly created USSR is Аэли́та [Aelita] made in 1924, directed by an accomplished filmmaker Yakov Protazanov and based on the novel under the same title by Alexey Tolstoy. It wouldn't be too much of an error to describe this movie as a Soviet version of the Edgar Rice Burroughs' Princess of Mars shot in a manner inspired by German impressionism. The story, much in line with the dominant ideology, focuses on the Terran expedition to Mars ruled by a local tyrant who is deposed during the popular revolution instigated by the visitors from Earth supported by the eponymous princess Aelita. The movie even incorporates an animated short Interplanetary Revolution made by Nikolai Khodatayev, Yuriy Myerkulov and Zenon Komissaryenko.
In the next year, the cinemas have shown another sci-fi movie, Aero NT-54 directed by Nikolai Pyetrov and telling a story on an ingenious inventor of a revolutionary aeroplane engine, prompting several intelligence agencies to steal the designs. Patriotic and distinctly anti-Western sentiments were also clearly seen in the movies sharing the main theme, directly inspired by the events of the freshly ended Great War and possibly made to raise the spirits after the lost Polish-Bolshevik War. These are e.g. Коммунит [Communit] by Yakov Morin or Наполеон Газ [Napoleon Gas] by Syemyon Timoshenko. Both movies focus on a usage of a novel chemical weapon in a war between USSR and the West with the former movie telling a story of an inventor working on such weapon to support the worker's revolution and the latter showing the heroic defense of Leningrad again insidious Westerners. As one can imagine, the propaganda content is relatively high in both pictures. Similar theme can be found in the 1925 movie Луч смерти [Death Ray] by Lev Kuleshov and Vsyevolod Pudovkin. It is worth noting that although the movies were mean to carry a political message, the enemy is depicted not as 'capitalist' or 'imperialist' but rather unnamed fascist countries in Europe and America, most likely modeled after the Mussolini's Italy. After a hiatus in the genre during the late 1920s and early 1930s, the sci-fi theme has been then picked up to a great effect in 1935, when Alexandr Andriyevskiy directed a movie Гибель сенсации [Loss of sensation], an adaptation of a 1920 novel Iron Riot by Volodymyr Vladko, directly inspired by the seminal novel R.U.R. by Czech writer Karel Capek (that is an origin of the word 'robot') and can even be considered a sequel of the latter. The other movie is Космический рейс [Space voyage] by Vasili Zhuravlyov that will set the theme for the next wave of the Soviet sci-fi after the Second World War. This movie, even though predates the space race, tells a tale of a first space travel to the moon and is a good example of an early hard science fiction set in outer space, made possible by the fact that a scientific consultant for this movie was no other that Konstantin Tsiolkovsky, commonly called a 'father of spaceflight' and creator of the theoretical models of rocket operation and movement in space.
The anti-Western themes will shortly return in the 1950s mainly represented by the 1953 movie Серебристая пыль [Silvery Dust] by Abram Room, telling the story of several foreign powers fighting over the eponymous 'dust', a new sort of weapon. These themes will be also present in the movie Торговец воздуха [Air Merchant] (1967) directed by Vladimir Ryabtsev, where a climatic crisis potentially endangering the life on Earth is being abused by a ruthless industrialist, although this is largely attributed to the fact that he plot is based on a 1929 novel by Alexandr Belyayev and thus explores the themes that were not common in the sci-fi literature and cinematography of the era. But after the Second World War ended with tense but relatively stable division of the spheres of influence and the position of USSR in the world, as well as similarly stable position of the new type of government in the country itself seemed to be finally established, the focus of the science-fiction books and movies shifted from the East-West antagonisms and the support for the global, or even local communist revolution to social issues. Туманность Андромеды [Andromeda Nebula] (1967) by Yevgeniy Sherstobitov, although still somewhat heavy on the ideological propaganda side, is based on the 1959 novel under the same name by Ivan Yefremov that has been considered by Boris Strugatsky a breakthrough in Soviet science-fiction. marking the abandonment of the 'short aim theory'. The latter, typical for the earlier Soviet sci-fi works portrayed a world few years or decades in the future, and its associated themes and tropes were largely derived from the contemporary issues. The new works focused on the far future, with the issues and possibilities brought by new developments being completely new and unavailable for the contemporaries (interstellar travel, real artificial intelligence etc.). Its portrayal of the communist utopia, not unlike one in the Noon. 22nd century cycle by Strugatsky Brothers also marks the shift of the focus from the technology itself to the social issues resulting from the scientific and technological progress.
Before Soviet filmmakers tackled the idea of space travel, they also turned their eye towards a closer, but also poorly explored area that were the ocean depths. Exploration of the oceans is the main theme of the movies Тайна вечной ночи [Mystery of the Eternal Night] (1955) by Dimitri Vasiliev and Abram Room and Тайна двух океанов [Mystery of Two Oceans] by Konstantin Pipinashvili (1957). Similar subject, although in a completely different setting is explored in the movie *Человек-амфибия [*Amphibian Man] (1961) by Vladimir Chebotaryov and Gennadiy Kazansky that is an early take on transhumanist love story. This last movie was a great hit in USSR (best grossing movie of 1962) and satellite countries.
As the space race and the expansion beyond Earth in general became headline news, such themes could not but find their way into a silver screen. The period between mid-1950s and mid-1960s in Soviet science-fiction was characterized by the works depicting the expeditions to extraterrestrial bodies to some extent mimicking the 19th century adventure and marine novels, focusing on the hardship and dangers early space explores must have taken. Among such movies we can find Планета бурь [Planet of Storms] (1962) directed by Pavel Klushantsev. In addition, Небо зовёт [The Heaven Beckons] made in 1959 by Alexandr Kozyr and Mikhail Karyukov anticipates tensions related to the space race that , as it tells the story of a race for Mars landing with the Soviet crew saving the American one that tried to get a headstart at the cost of inadequate preparation.
It is worth noting that although Soviet cinematography was generally not distributed or even known outside the USSR, with some movies being sometimes screened in Eastern block countries, several movies were bought by American distributors and after heavy editing were screened abroad under different titles. This is the story behind Небо зовёт that has been edited by Roger Corman and screened in USA as Battle Beyond the Sun and the aforementioned Planet of Storms was edited so thoroughly that it has been turned into two separate movies: Journey to the Prehistoric Planet (1965) and Journey to the Planet of Prehistoric Women (1968). It is worth noting that the editing and production of these involves several high-profile filmmakers, such as Francis Ford Coppola or Peter Bogdanovich, although they generally were hiding behind pseudonyms (Thomas Colchart and Derek Thomas, respectively).
A more humorous take on the development of technology, similar to the one present on the short stories of Robert Sheckley can be found in the movie Его звали Роберт [His Name was Robert] made in 1967 by Ilya Olshvanger, where an eponymous android, non-recognizable from a human and developed for the operations in environment lethal to humans (not unlike Dick's replicants from 'Do Androids dream of Electric Sheep') falls in love and ultimately breaks trying to perform every task he is asked for. Although the movie is a light-hearted comedy, it nevertheless explores many serious facets of the practical applications of the artificial intelligence. Кин-дза-дза! [Kin-dza-dza!] (1986) directed by Georgiy Danelya is an interesting example of a Soviet movie that keeps in line with what one would expect of such a work, but also introduces surrealist elements that, complete with a distinctive 'raggedy' aesthetics that evokes movies of Alex de la Iglesia, especially Accion Mutante (1993). Possibly due to the strong anti-establishment undertones and haphazard imagery and plot, the movie still has a cult following, with its animated version being made recently.
Some useful links:
What was genre fiction like in the Soviet Union? with an answer by u/Noble_Devil_Boruta
Was there any fantasy and science fiction in the USSR?, with an answer by u/Pseudohistorian
A huge answer on nuclear apocalypse and Soviet science fiction by u/lkieslowskifan