Entering Bulgaria is no different to entering any other EU country, such as France or Spain. The only time you need to worry about money at the border is if you're carrying more than (the equivalent of) �10,000, which you must then declare.
I have travelled overland through Bulgaria and into Turkey and did not offer money at any time to border guards. I had no trouble getting through. You have to pay a few quid for a visa going into Turkey but you just go to the appropriate hatch and pay the bloke there, after getting your passport stamped.