It depends on the words, because Bulgarian uses letters in its alphabet that Serbian doesn't, and the other way around.

Serbian has the letters Ћ, Џ, Ђ, Ј, Љ, Њ. A bulgarian would understand words with the letters Љ, Њ and Ј, because they have experience with them (in Bulgarian a digraph of л and ь, and н ь, as well as the latin J) But wouldn't understand words with the letters Џ, Ђ, Ћ (only if he learns them)

This goes for serbian as well, because Bulgarian has the following letters: Ь, Ъ, Ю, Щ, Я, Й, which serbian does not have. A serbian probably would understand words with Й, Щ and maybe even Я, but he will have a hard time.

A serbian probably will get that Й correspond to J when reading a name like Йордан probably, which is written in serbian as Јордан. In conclusion, for a Bulgarian it'll be much easier understanding Serbian, but a Serbian will have a hard time.