There are two term that are sort of competing with each other so far - which if them will become de-facto standard.
One is "картонный подстаканник". The other one is - "капхолдер". There's also such thing like literal translation - "манжет на стаканчик" or "манжет для стаканчика" but de-facto it is less used than first two terms.
Surprisingly, though the second term is direct borrowing from English, in English cupholder is slightly different thing.
Here's what Russian retailers sees as cupholders:
As a side note - this phenomenon is not that rare, it is called pseudo-anglicism and there's a bunch of other neat examples - check out this video for instance (apart from the author mocking pronunciation - it's actually norma that words are phonetically changing).