you are very adventures. There is a bus, costs next to nothing, no need to prebook. Hard to explain where to catch it, I might not find it myself. You can get a taxi and ask it to take you to the bus station. You will be the centre of attention there, being foreign. Perhaps some taxi drivers will even offer you a trip to Bansko. If you are not alone, it might be worth the money.
Why not rent a car?
Are you from Austria or Australia?
The bus sounds like a challenge and we are very strapped for cash so I am going to give it a go. There are two of us so hopefully we should work it out.
Bus costs 7-8 leva just under 3 GBP takes 3 -31/2 hours as it only stops twice so the journey is not much longer than a taxi, taxi would cost you 140 leva each way
you would need to get a taxi from the airport to the bus station, agree taxi fare prior to getting in cab should be 10 -20 leva
buses run just about every hour (my son and husband have done the journey)
Bus station is at the bottom end of town (you need to walk uphill from the bus station)
hope that helps