The city center is not that great I'd recommend  Lozenetz or Vitosha. In the past year on three occasions I had friends looking for apartments and I also went trough this about two years ago.

There is only one thing to it: use real estate agency. About 98% of the ads for apartments for rent are from real estate agencies even the ones that appear to be from the landlord. And If you just try calling, the agent will play stupid and loose your time until you sign a contract with his agency( it may take for about a week or two).

Just find some agency directly and take some bulgarian to look very well trough their contract and make sure you give nothing upfront and their commission is no more than half the rent of the apartment they find. And the whole thing takes about two weeks.

For about 550-600 you can find great furnished apartment.