There are many options: One is to go to ThaBo and then take the 2020 to Ban Phue or take the 2266 in Si Chiangmai and near Klang Yai take the 2348 to Ban Phue. This route will bring you past the Phu Phrabat Historical Park (quite nice but can be quite hot during the daytime).
From Ban Phue again several options. Easiest is probably to go to Kut Chap either taking the 2021 and in Kham Bong take the 4018 south and then in Kut Chap take the 2314 and after some distance turn right (southwest) to Nong Saeng and then the 3018 to the 210 and west on to NBL. When you reach the 210 there is nearby the Hoi Hin Lan Pi National Park (very old fossils, etc.) which may be worthwhile (is on my list of things to see).
The other option from Ban Phue is to take the 2098/2097 south to Kao Kloi and then head easterly to Kut Chick on the 2004 and the 4016 to Nong Phai Sun and then on the 210 east to NBL.
I took a baht bus once from Nongkhai to Na Klang via Tha Bo and Sri Chiang Mai . I seem to remember it's a straight run south from Si Chang Mai through Ban Phu down to Na Klang (209?) no mountains though:-( then a quick squirt down 210 to Nong Bua.