Traveling from Minamata to Himeji, you have the option to take the Limited Express train service known as the 'Nichirin'. This train offers comfortable seating and makes the journey between the two cities in approximately 4 hours, with speeds reaching up to 120 km/h (75 mph). The route is scenic but includes several stops along the way, giving passengers a chance to see various towns and landscapes of Kumamoto and Hyogo prefectures.
For those looking for a faster option, you might consider transferring to a Shinkansen service at Shin-Yatsushiro Station. From there, you can catch the Hikari or Mizuho Shinkansen, which can cover the distance to Himeji in about 1 hour and 15 minutes, reaching speeds of up to 285 km/h (177 mph). However, this option requires the additional transfer and may not be considered direct travel.
Overall, the Nichirin service combines a leisurely pace with a solid glimpse into the region, while transfer to the Shinkansen offers a much quicker, though more complex, travel experience.