There is a taxi company in Udon. I have used them several times and they are quite cheap. However, it has always been the bar owner who has called for me so no idea on the numbers.
There is a shuttle bus service to downtown Udon. Not sure where it goes but not too expensive. I think it used to be 80 Baht. Also a couple of years ago after arriving at the airport I asked about a taxi and there was a couple of "independent" drivers outside and we got a ride to our house for a couple hundred Baht.
if u arrive at udon airport.when u walk through the door into public area.on your right hand side ist little office.they have a sighn saying taxi. good and cheap.the prices are on the wall.udon 80bht.i use them a lot.if u go outside you will get ripped off.its the same office where you can leave your car.secure compound.
http://www.ud-carrent.com has their own taxi service for both in Udon Thani and outside. You can make a appointment to get picked up right away from the airport arrival hall or from your home/hotel address
I think if they have room, and for a small fee the Charonsei Hotel (Centara as its now know) shuttle mini bus will take you to central udon. Use this as a back up plan maybe ?