Takachiho itinerary and Transport Questions

Hi all,

I will be based on Kumamoto and is planning to take a day trip to visit Takachiho Gorge & Amano Iwato Shrine.

I have 2 plans and would love to get opinion on which is better and also a few more questions about transport in that area.

Plan 1
09:11 - Bus from Kumamoto
12:19 - Arrive at Miyakoh Bus Center
12:37 - Take bus to Amano Iwato Shrine
*Plan to visit the West Shrine & Iwato Cave. If time permits, grab a quick lunch around that rea

14:13 - Take bus back to Miyakoh Bus Center ( is it possible to get a taxi from Amano Iwato Shrine to Takachiho Gorge directly?)
*get late lunch if didn't manage to have it earlier

15:00 - Takachiho Gorge
16:57 - Bus from Miyakoh Bus Center back to Kumamoto


Plan 2
09:11 - Bus from Kumamoto
12:19 - Arrive at Miyakoh Bus Center
12:30 - Takachiho Gorge & Lunch
15:00 - Take taxi to Amano Iwato Shrine (is it diffcult to get a taxi here?)
16:10 - Take bus back to Miyakoh Bus Center
16:57 - Bus from Miyakoh Bus Center back to Kumamoto

A few other questions I have:
1. Is Go Taxi app available in this region?
2. How hard will it be to get a taxi around here?

Any food recommendation?

Thanks in advance.