According to their web site, this hotel does not have public mixed onsen. Instead, they have private open-air bathes for couples and families (15:00 - 21:00, 40 min, 2,160 yen, hotel guests only). I would recommend to book in advance if you would like to try the open-air bath.

Hi there, I drove a car from Kobe. So my answer is not on my own experience but on my web search results. Sorry I cannot refer useful sites here because of the posting guideline. I would say contacting to Miyoshi City Tourist Information Center might be helpful. I have check out the hotel official Japanese web site. They said there is no complimentary shuttle service from Oboke station. They just say "by taxi 15 min, 3000 yen" There is local bus service by Shikoku Kotsu from the station to the hotel (Soya line, about 30 min). Please noted it is not free (about $5) and service is not frequently. There is one rental car service at Hotel Obokekyo Mannaka and another one in Awa Ikeda JR station. Good luck and enjoy your trip in Japan.

we were on.a walking tour sorry cannot help with directions