May 18- May 31: Itinerary check and questions for 13 days in Japan: Tokyo / Kyoto / Fujikawaguchiko / Nara buddhist temple stay / Pit stop at Eiho-ji temple near Nagoya

Hello, we are a 32F, 33M couple super excited for our first trip in Japan. We are both in the design field, enjoy architecture and nature. I’ve more or less finalised our Itinerary- grateful for any suggestions on this:
**Day 1 (May 18): Tokyo, Asakusa**
Land in Tokyo at 5.30am, start the day by 10am
Sensoji temple
Asakusa tourist info centre
Chidorigafuchi
Isseido Booksellers
Yasukuni Shrine (based on time)
**Day 2 (May 19): Tokyo, Uneo**
Ameyoko shopping district
Uneo park
National Museum of Western Art (corbusier)
National museum of nature and science
Tokyo national museum
Nezu shrine
**Day 3 (May 20): Tokyo, Chiyoda**
Imperial Palace East Garden
Imperial palace
Tokyo international forum
Art aquarium museum
teamLab Planets TOKYO
Hama Rikyu Gardens Pier
Imperial Hotel Tokyo (based on time)
**Day 4 (May 21): Tokyo to Fujikawaguchiko**
Train from Tokyo to Fujikawaguchiko
Saiko Iyashi-no-Sato Nenba (based on time)
Fuji Shiba-sakura Festival Observation Deck
Oishi Park
Yamanakako Onsen Benifuji no Yu hot spring (based on time)
**Day 5 (May 22): Fujikawaguchiko to Tokyo Shinjuku city**
Spend the morning biking around Fujikawaguchiko and take train back to tokyo at 3pm
Tokyo plaza omotesando
Omoide Yokochō
Shinjuku golden Gai
DIOR Tokyo Omotesando
**Day 6 (May 23): Tokyo, Shibuya**
Tokyo metropolitan gov building
Shinjuku Gyoen National Garden
Shibuya station
Meguro River Cherry Blossoms Promenade
Nezu Museum
**Day 7 (May 24): Tokyo to Nagoya, Nagoya to Kyoto**
Train to Nagoya, stop at Eiho-ji temple
Reach Kyoto around 6pm
Walk in Goin
Menbaka Fire Ramen
**Day 8 (May 25): Kyoto**
Sagano Scenic Railway
Arashiyama
Gio-ji
Adashino Nenbutsu-ji Temple
Kosanji
Fukuda Art Museum
**Day 9 (May 26): Nara / stay at buddhist temple**
Checkout of hotel, train to nara, leave bags at train station
Nara park
Isui-en Garden and Neiraku Museum of Art
walk around Naramachi
Todai-ji Temple,Kasuga-taisha Shrine, Horyu-ji Temple, Yakushi-ji Temple
Cab to Gyokuzoin (Buddhist temple)
Arrive by 5.30 for dinner by 6
**Day 10 (May 27): Nara to Kyoto**
Early morning prayers at 5.30, breakfast
Unsure of what else to see here
Train back to Kyoto
Jingo-ji
Kosanji
Ryoan-ji Hojo Teien
**Day 11 (May 28): Kyoto**
Kinkaku-ji
Daitoku-ji Daisen-in
Nijō Castle
Kōdai-ji Temple
**Day 12 (May 29): Kyoto**
Saihō-ji (Koke-dera) Temple
Jizo-in
Katsura Imperial Villa
Bamboo Path (Take-no-michi)
Kōdai-ji Temple
**Day 13 (May 30): Kyoto / shopping / day trip**
Shopping if possible
Day trip to Uji / Osaka / Ine / Kinosaki Onsen
**Day 14:**
Flight back from Osaka
Some questions about the itinerary:
1. I’m not really sure about the pit stop at Eijo-ji temple- it looks beautiful but we lose a good half day getting there.
2. Still trying in vain to find a public Rotenburos in any of the locations
3. Still trying in vain to find a private onsens that can be rented for a couple of hours
4. I’m not really sure of the stay at the buddhist temple- we’ll have to wake up at 4am to experience the fire prayer, and that would probably destroy us with the added jetlag
5. I’m a vegetarian,my partner is not- so any restaurant recommendations would be hugely appreciated.
Many thanks in advance for taking the time to read through this long ass post