You can see Rila Monastery in a day trip, or you may even sleep there. I'd suggest not to rely on public transport, better take a tour. There are a lot of places for hiking in the mountains, but December is not the time for this.
The roads are best avoided if it has just snowed, but it does not happen too often in December.
I am a Bulgarian and may help with info, but have in mind I don't live there now (just visit once or twice per year).