Fly to Khon Kaen - trust me this is the best option. I've done it a few times. Very easy, KK airport is very comfortable. Grab a room and wait for the bus that leaves in the middle of the night (there is only one bus per day from KK to Mukdahan and it is extremely comfortable). DON'T take the minivan that leaves every couple of hours. The vans are in very bad condition, the drivers drive very dangerously, A/C doesn't work, seats are oversold, etc. and the drive from KK to Mukdahan is by far the worst I have ever encountered anywhere in Thailand. At least on the bus you can sleep.