I assume you refer to Takachiho.You won't be using expressways, at all. They are all toll-free roads.
I recommend to visit Takachiho before Kurokawa:
From Kumamoto take National Route 57 to Mount Aso, then Route 325 to Takachiho.
After Takachiho, backtrack to Mount Aso and select any road over the northern rim of the caldera to Kurokawa Onsen, e.g. National Route 212 or Prefectural Route 11.
From Kurokawa, I recommend to follow the Yamanami Highway (Prefectural Route 11) via Yufuin to Beppu.