Older people have learnt Russian at school to start with (even in places such as Hungary or East Germany). Slavic languages do have similarities, and you’ll find that if you’re fluent enough in one of them, you’ll understand quite a bit in others. But younger people may not necessarily understand you (or want to).
There are also social reasons. For example, in Poland it’s a lot easier to communicate in English than in Russian, but the reason for that is more political than linguistic.
Overall though, I’ve found that my few words of Russian and my basic Czech meant that I at least understood some basic words wherever I went and was able to pick up the local language a lot faster.
(btw only Bulgaria still uses Cyrillic as far as I remember).