Indeed they connect with Chiang Mai twice a week and Bangkok’s Don Mueang Airport no less than four times per day. The Chiang Mai flight, run by Kan Air, departs Fridays and Sundays at 14:50 and costs around 1,890 baht while the Bangkok planes leave at 08:45, 10:35, 12:45 and 17:05 with prices varying widely between 1,500 and 3,000 depending upon time and date. These are ATR prop planes so allow an hour for Chiang Mai and 1.5 hours for Bangkok. Times seem to vary too so do please check beforehand.
Nan’s tiny airport is located just to the north of the town centre and its code is NNT. Airport taxis will set you back 100 baht to any downtown hotel.