Transit through Bulgaria with Schengen visa?

I have a Schengen visa type D+C, with a 1-year permit of stay. I am flying to Bulgaria (EU member, no Schengen agreement) and would like to transit to Serbia (the same day). Will I have any problems doing so? The permit of stay I own allows me to travel to Switzerland for example (non-EU, non-Schengen)... Anyone knows?