The website states that metro stations sell a 3-day smartcard ticket, but the staff member said this was only available at the city centre transport headquarters.
I did manage to buy the 72 hour ticket from the airport metro ticket office. It took a while, the lady didn't speak any English. But eventually got it, on the Ultralight card rather than the plastic Sofia City Card that I really wanted.
BGN 15 for the 72 hour ticket plus BGN 0.80 for the smart card itself. The plastic card would be BGN 3 if you can get it.
The ticket office is still cash only, but there is an ATM right next to it, which didn't levy any stupid fees or DCC (for me anyway - used a Wise card).