There's a bus service from Aso to Takachiho but all the online information is available in Japanese only.
The bus information can be found here:
Here is the time table for week days:
The bus starts at Kumamoto [熊本駅前] then arrives at Aso [阿蘇] and Takachiho bus center [高千穂バスセンタ]. There are only two buses per day, so planning is key.
There are direct buses back to Hakata or Kumamoto (from where you can use the Shinkansen to Hakata). Looking at the amount of travel involved, I would not visit both in a single day trip.
Hope this helps,