Wikitravel has a section on this.
Essentially you'll want to learn the script - even if you can't speak it. It at least helps you say words, or if someone says to try the cheese and tells you the word is 'Kashkaval', at least when you see the script saying 'Кашкавал' you'll be able to figure it out, even if slowly. It's immensely useful for street signs and being able to order off a menu - or even to learn what the name for something is on a menu if it has pictures (I'm speaking especially from experience with the Cyrillic script in Russia etc).
Often you will understand it anyway, as it sounds like the English or French word.
In terms of being understood, however, in the cities it should be easier. As Wikitravel says:
...as Bulgarian education emphasizes foreign language studies,
especially English language, it wouldn't be a problem to talk and find
information in English in bigger cities. It's best to turn to the
young population for a direction or an advice.
As it's related to Serbian, Russian and other Eastern European languages, if you don't want to learn Bulgarian, perhaps you could consider one of those. I'd suggest Russian, as it would help you in several other countries in the area.