A day or festival dedicated to a town or city

Solution 1:

I think in context "Mtsensk Day" would be both clear and natural. It's not a concept I'm aware of in any English-speaking town, but we do have Yorkshire Day for the county.

Solution 2:

The equivalent in English-speaking areas are American county fairs, British county shows, and (patron-)saint's days. Municipal festivals as such are less of a done thing, although some cities have favorites—e.g. New York's St Patrick's Day & Halloween blow-outs—and Canada apparently has a bizarre generic 'civic holiday' on the first Monday in August. It's all on the same day across the country, but some municipalities rename it for their own observances, calling it (e.g.) "John Galt Day" in Guelph, Ontario.

You could gloss the Russian idea as "a local" or "town festival" but using the word "fair" runs the risk of confusing it with traditional regional markets and "show" obviously usually implies that there's just one longish performance. You're probably best off going with Mtsensk Day and spending a sentence, parenthetical, or footnote explaining the Russian practice.