It must be this shrine, near Aomori city
https://maps.app.goo.gl/rWPJUE5CSW5wnFt99You can rent a car for one day if you want to visit it. I don't think there is any discount for renting a car for multiple days.
Although I haven't been to most of the places you listed, it seems to me that your day 9 is going to be technically impossible. I rearranged your trip to use 2x Tohoku area pass, which will be less expensive than a national JR pass. I also removed one night from Ginzan onsen (been there, one afternoon and morning was more than enough unless you are planning to visit outside the town).
(Tohoku pass #1)
Day 1: Arrive Narita in the afternoon, JR to Fukushima
Day 2: Tsuruga Castle, Aizu Bukeyashiki, Ouichi-juku
Day 3: Kokeshi in Shiroishi City Miyagi Prefecture (not sure this is doable), train to Sendai
Day 4: Geibiki gorge (check the cruise timings), Pokemon Joyful Train
Day 5: Yamadera (early morning), Ginzan Onsen Town
Day 6: Somaro Maiko House, Kamo Aquarium (not sure you'll have time),
rent a car to a shukuboDay 7: Dewa Sanzan (climb Yudono, Haguro-san)
Day 8: Dewa Sanzan (climb Gas-san), Churenji Dainichibo
(Tohoku pass #2)
Day 9:
drop off the car, train from Tsuruoka to Kakunodate
Day 10: Hanamaki Onsen, Wanko Soba at Hatsukoma
Day 11: Aomori City, Inakadate Tanbo, Nebuta Museum
Day 12: Takayama Inari shrine (rent a car if needed), night in Aomori (?)
Day 13: Matsumoto City Museum of Art or Tokyo
Day 14: Tokyo
Day 15 - 16: Tokyo
Day 17: PM flight from Haneda