Navigating Saga prefecture

Hello I would like help here. I intend to visit from Fukuoka-Yoshinogari historical park-Yutoku Inari Shrine-Ureshino Onsen within a day using SUNQ pass. The JR pass is easier but it does not cover the Mizuho train from Fukuoka-Yoshinogari during transit (upon checking Hyperdia) and I am on budget. From Yoshinogari park however, I am wondering which buses to take to Yutoku Inari shrine.

Sorry, I can't answer your question specifically. Your best bet is ask Fukuoka Tourist Information for bus info since I presume you will be entering Kyushu there. You can probably ask at the park too.

But I highly applaud your itinerary. The Jomon theme park is highly recommended. It is one of the most well kept I have seen in Japan, and the elevated structures are quite interesting. You can also sample some Saga beef there for quite cheap. They offer a 200 gm steak teishoku (set lunch) for around 3300 yen at the restaurant near the entrance.

The Yutoku Inari shrine is much underrated because it is out-of-the way. It is very picturesque and colorful. Puts Kiyomizudera to shame in this aspect. If it was located near a city like Kyoto, it would receive much more press.

For those who are interested, a better way to visit the shrine is on a rental car trip around the coastline of the Ariake Sea, from Shimabara to Kumamoto. It's the only place in Japan with mud flats like Korea or China. Sample deep fried mud skippers. The huge Takezaki crab in the area is not to be missed, and are quite pricey (expect to spend 6,000-10,000 yen for a teishoku). If you haven't been to Yanagawa before, you can cover that in a short side trip from the coast.

Hakata to Yoshinogari-Koen is doable without using the Mizuho Shinkansen, by using instead the Sakura Shinkansen or the Kamome Limited Express. You might need to untick the "Nozomi/Mizuho" box in Hyperdia to see those options.

Also, going from Yoshinogari Park to Yutoku-Inri by bus seems complicated. Your best option is probaly to take a train from Yoshinogati-Koen to Hizenhama, and then the shrine is about a 30 minute walk (or 15 minutes by bus, but there is one connection so you might lose some time).