The first question would be whether these strings use a distinct Unicode character for this purpose, or a simple straight quote.

Assumimg the latter, can you post a sample or two to show the positions in which primes appear? I'll be curious whether they appear with space before, space after, no surrounding space, and in the characters immediately adjacent.

Too early to say for certain, but it is possible that a clean solution to this will not be possible in Zotero. The Jurism variant of Zotero supports language- and script-tagging of individual fields, something might be done there.

Thank you for your comments. Here is more information:
1) I am not exactly sure whether publishers use a prime (U+2032) or a straight quote, but the explanation on the character that is used in Russian transliterated texts calls the character "a prime": http://homes.chass.utoronto.ca/~tarn/courses/translit-table.html
But it seems that either (a prime or a straight single quote) should look fine in transliterated names/titles from Cyrillic - I am not a publisher, and both characters (a prime (U+2032) and a straight quote) look pretty similar to me.

2) Given that a prime substitutes a particular letter in the Cyrillic alphabet (the soft sign - "ь"; and a double prime - the hard sign - "ъ", ugh, I know), the placement of this character can be literally anywhere in the transliterated text, except for the beginning of the word; and a space may or may not follow this character (but a prime is never preceded by a space). Please see some examples here: https://drive.google.com/drive/folders/17lfbJg5v27k1Lf75ae5QWrDe-pPt3e6R?usp=sharing
In all these texts a prime or a straight quote (i.e., the desirable look) is highlighted in yellow and what Zotero does is highlighted in green.

I might be wrong, but it seems that it should be possible to make some changes in Zotero which would allow not to substitute single or double quotes with "curly" ones, if the source is marked as Russian (or any other language that uses Cyrillic) in the database. After all, the issue with capitalization of titles works perfectly based on the language of the source (the rules of title capitalization in Slavic languages are also different--there, only first world in the title plus proper names are capitalized). And titles for Slavic sources in Zotero have been capitalized (or rather, not capitalized, heh) correctly, if the source is marked as "Language: Russian" (or any other Slavic language) in the database. This makes me think that there might be a possibility to retain straight quotes for only those sources that are marked as "Slavic languages." Many thanks!

I can look into this if there is some test data to work from. To submit data, join this public library, sync it locally, drag a few (two or three) relevant entries into it, and either illustrate or explain the correct result in the Abstract field or a note attachment. Sync the items up to zotero.org, and I'll take a look when I'm able.