"I’ve never heard of any instances of Russia and the Italians warring in the Second World War"

It is relatively under-discussed, even in histories of Italy during the Second World War, but Italy was massively involved militarily on the Eastern Front, mostly between 1941 and 1943. A third of Italian combat losses in the war (which was about 230,000 on all fronts) between 1940 and 1943 were there fighting against the Soviets, and 229,000 personnel were involved in actions on the Don in 1942-1943, or five times the number of Italians engaged at El Alamein during the same time period. The Italian Navy was also involved in operations in the Black and even the Baltic Sea during this period. Probably the most infamous role the Italians played were that the Italian 8th Army, covering the left flank of Stalingrad, was attacked by the Soviets in Operation Uranus, which was the offensive that began in November 1942 and successfully isolated German forces in Stalingrad. Something like 60,000 Italian POWs were taken by the Soviets, a significant number of whom (in the tens of thousands) died in captivity and most of the rest returning to Italy after the war. Interestingly in 1970 a Soviet-Italian co-production released a film starring Sophia Loren and Marcello Mastroianni that dealt with the fate of Italian soldiers and POWs in the USSR.

Anyway, as for what happened with the Italian fleet as a whole: when the Italian government deposed Mussolini and accepted armistice terms from the Allies in September 1943, part of those terms stipulated that the Italian Navy sail to British-held Malta to be interned. The battleship Roma was in fact sunk by German radio-controlled bombs with a massive loss of life as part of this operation, but most of the fleet (including Giulio Cesare) made it to Malta. While most of the Italian Navy was allowed to operate later as "Co-Belligerents" under the Italian royal government based in Southern Italy, the battleships were interned in Egypt for the remainder of the war.

Italy signed the Paris Peace Treaty in 1947 - if anyone is so inclined, they can read the full English text here. This treaty actually allowed Italy to keep a navy, but limited its total displacement to 67,500 tons. Resrictions were also in place forbidding Italy from constructing, acquiring or replacing battleships, aircraft carriers, submarines, torpedo boats or assault craft, as well as other limitations on naval construction until 1950.

On top of this, Italy agreed to pay reparations to specific Allied countries: Yugoslavia was to get $125 million, Greece $105 million, the USSR $100 million, Ethiopia $25 million and Albania $5 million. In the USSR's case, the reparations were able to be paid in factory equipment (especially those related to war material production), Italian assets in Eastern Europe, and Italian goods as determined in Soviet-Italian negotiations (subject to approval by the US, UK and France).

Italy ended the war with six battleships intact: the Andrea Doria and Duilio, the Giulio Cesare, and the Littorio and Vittorio Veneto. The first two (which were the oldest) Italy kept and operated in the postwar era, Giulio Cesare was transferred to the USSR as part of the reparations agreements, and Littorio and Vittorio Veneto were transferred to the US and UK respectively under reparations claims allowed under the Paris Treaty. As it happened, the US and UK disposed of their battleships very quickly, with them being sold and broken up for scrap in Italy almost immediately. What was unique isn't that the Soviets got an Italian battleship, but actually bothered to use it.

There were practical reasons for this. The Soviet Union and Fascist Italy had a history in the 1920s and 1930s whereby the Soviets purchased Italian naval designs and even a few ships (notably the destroyer Tashkent). In the late 1930s in particular, Stalin was looking to get modern battleships as soon as possible, either through the purchasing of plans or of ships built outright from Italy or the US (more background on that is here). As things worked out, the Soviets began construction on four Sovetsky Soyuz class battleships based on the design of Littorio, but these were never completed before the war, and what had been built of the four was scrapped after the war. Nevertheless quite a few Soviet warships had Italian-designed systems in them (as u/kieslowskifan describes in great detail here) and so Giulio Cesare, now renamed Novorossiysk and transferred to Soviet service in 1949, was used for training purposes, and would have had genuine value as a compare-and-contrast between Soviet naval systems based on Italian designs, and the genuine article. As it happened, the battleship was sunk, most likely by a German mine, off Sebastopol ten months after the transfer (there are lots of conspiracy theories swirling around this).