Did anybody ever succeed using the Unicode combining acute accent with cyrillic text in InDesign? If so, I'd like to know about your OS and InDesign version.
Thank you for your interest.
Check out this donation-ware script from Peter Kahrel: InDesign: Enter characters with diacritics and other special sorts | Peter Kahrel .
"The script lets you enter all kinds of combinations of letters and diacritics, even when the combination doesn't exist in the current font."
I have never had any need to do so, as I have a personal loathing for combining accents (having been burned in the past in Vietnamese projects) but I can test it pretty easily. It works without issue on CC 2015 on OS X Yosemite. I can test on a variety of versions and platforms, if you want to know something specific. I would suggest you tell us what fonts we should use, too.
If you need Cyrillic with accents for some non-Russian language, there are a few different ways to go. I once again advise against combining accents - you can always use the fine IndyFont tool to cook up a single precomposed glyph that is properly encoded and doesn't risk dropping a combining accent when you switch to a font that doesn't have the combining acute.