There is archeological evidence that horses have been in Japan for thousands of years, however the earliest records of Japan by the Chinese in the third century say that Japan did not have any horses. It is known that horses were introduced from Korea in the 4-5th centuries and this seems to be from when most Japanese horses are descended. The above manhole is from Mochizuki, a small town in Nagano. The area bred and raised horses for the Imperial court since ancient times.
Kushima, at the southern tip of Miyazaki, is home to a breed of wild pony, the Misaki Pony, considered a Japanese breed. They live on Cape Toi and are a tourist attraction. there are about 100 of them. Misaki means "cape".
This final one is from Ishigaki Island in Okinawa, and other than the fact that the area offers horse riding as an attraction I can find no explanation for the horse on their manhole design.