As has been already pointed out by a few other posters, your current itinerary sounds hectic.
Regarding your Nara trip
: Horyuji Temple and Todaiji Temple are located in quite different directions of JR Nara and Kintetsu-nara Stations.
: Yoshikien Garden, Isuien Garden, Todaiji temple and Kofukuji Temple are located rather close to each other.
: Within several hours you may travel at a high pace to visit these four sites, maybe plus Naramachi, but your time may well run out if you add Horyuji Temple.
: Why don't you plan to visit Kasuga Taisha Shrine?
- Nara Kotsu Bus: Bus passes:
http://www.narakotsu.co.jp/language/en/free-ticket/index.htmlRegarding your trip to see Mount Fuji
: You need about 4 to 5 hours just to travel from Shinjuku to Kawaguchiko and back.
: It sounds non-realistic to expect only 3 hours or less for you to tour the Fuji Five Lakes area.
: On your way to Tokyo, you can make a side trip from Odawara to Hakone, where you can have views of Mount Fuji unless the weather is too bad.
Regarding transportation to Tokyo
: A Shinkansen train does not bring you to Shinjuku Station.
: In Shinagawa Station you can make a transfer from Tokaido Shinkansen to Yamanote Line (outer loop), passing through Shinkansen South or North Transfer Gate.
https://www.jreast.co.jp/e/stations/e788.htmlRegarding your Tokyo trips
: The statue of Hachiko dog is near JR Shibuya Station, not near JR Shinjuku Station.
: Spots called as Shinjuku something are studded in a very wide area, which stretches over the boundary between Shinjuku and Shibuya Cities.
: You had better narrow down sites in Tokyo so that you can visit them in a day with enough time allotted for each of them.
If I were in your shoes, I would discard one temple and one garden in Nara and two gardens in Tokyo so that I can have much time for my trip to see Mount Fuji.