Russian tourists ususally get away without knowing Bulgarian because of the similarity of the language and they pick up Bulgarian quickly. Pronunciation is very easy for Russians. Bulgarians are also able to understand a lot of Russian, even if they never learned it but you can't rely on it. Bulgarian and Russian are close but there are also a lot of differences.
As Bulgarian like Russians are Christian Orthodox, written Russian borrowed a lot from Bulgarian as old-Slavic roots, which can't be said about Ukrainian or Belarusian (they borrowed more from Polish). A few examples of old Slavic roots borrowed from Bulgarian (there are also East Slavic roots for each of them): врата, враг, злато (злато-серебро), брада (брадобрей), град (Волгоград).
I never learned Bulgarian but if I open a Bulgarian text, it makes sense to me, I can understand quite a lot. It would be hard to follow the news on TV, maybe, although I would understand 30% to 70% but daily conversations would be quite easy to understand.
The reverse (Russian understood in Bulgaria) will be true because were more exposed to Russian than Russians to Bulgarian. They would have more trouble with the Russian grammar - complex case system is non-existent in Bulgarian (and Macedonian) but the have a notion of a definite article.