Many Udon Thani hotels provide airport transfers. Check with your hotel to confirm availability.
Taxi to Udon Thani city center is 60 baht.
There are four daily buses direct to the Friendship Bridge in Nong Khai from the airport, roughly synchronized with flight arrivals. Buy tickets (150 baht) at the taxi desk. Note that these buses will drop you off into a pit of tuk-tuks, which will try to convince you that you need their services at 20 baht a shot; unless you have baggage, you can ignore them and just walk out to the road, from where you can see the Thai checkpoint less than 500 metres away.