Just found this for you:
"ATMs/Cash Machines (the Bulgarian word for them is Bankomat) are widely available. There are two at the airport - one in the departures hall and one in front of the Bulgarian customs building, which is outside and to the left of arrivals. Strangely there are no ATMs in the arrivals hall."
source: http://www.local-life.com/sofia/guide/money

And:
"ATMs / Cash Machines - Yes, located in the Arrivals and Departures Halls (landside)." source:http://www.sleepinginairports.net/europe/sofia.htm

I don't recall specifically about the Sofia Airport's ATM, but whenever I arrive in a new country I locate and use an ATM right after exiting from immigration/customs. I probably asked someone and they pointed me to where the ATM was.