When Ukraine had its much-interrupted first period of independence in 1918-1920, it introduced new postmarks which were in Ukrainian, not Russian. My Russian is terrible and my Ukrainian nonexistent, but you can usually spot them simply because the spelling is different from what you're used to, e.g. Kiiv instead of Kiev, Kharkiv instead of Kharkov, etc. A few of them lasted well beyond Ukrainian independence into the mid-late 1920s.
But - and here's where it gets interesting - when many of the Union Republics that constituted the USSR started issuing new postmarks during the 1920s, the postmarks were bilingual: Russian and the local language. So yes, the Ukrainian SSR introduced many of these bilingual postmarks during the 1920s. However, there were some places (not many) that introduced all-Ukrainian postmarks, and apparently Tsibuliv (Tsybulevo in pre-Revolutionary times) in Kiev Oblast' was one of them.
They're worth looking out for, but count on seeing them very rarely. And by the way, the whole Ukrainian SSR period is almost a white spot on the philatelic map: apart from the "Index" postal code introduced in the 1930s I don't know of any subjects from this period that have been studied, let alone exhibited.
It's a pity because Ukraine was actually a very interesting place, politically. The conscious policy of korenizatsiya or "indiginization" led to growing national consciousness as Ukrainian communists tried to develop a separate national identity within the USSR. There's a great book on the subject, James Mace's "Communism and the dilemmas of national liberation" that follows the process and its abrupt end in the early 1930s. I wonder if the maker of this Tsibuliv postmark was charged with nationalist deviations during the Purges....
