What time of the year are you travelling? This could make a big difference for nature places.
Based on the below itinerary, would I need to a Jrail pass?
Doesn't look like a JR pass would make sense. Perhaps one of the regional rail passes could pay off.
3 DAY TRIP TO IBARAKI, FUKURODA FALL, KAIRAKUEN PARK, MT TSUKUBA
Not sure, but this might be one sight too much for one day.
4 TSUKIJI, MEIJI SHRINE, SHINJUKU GARDEN, RIKUGIEN, KOISHIKAWA KORAKUEN
I would not bother with Shinjuku Garden ("gyoen") which is mostly just a regular western style park. It is a good place for a picnic, but with little time in Japan I would focus on more Japanese things. Rikugien and Koishikawa Korakuen are fine gardens for Tokyo but you will see equally good and better in Kyoto. Same for Meiji Shrine and Asakusa/Sensoji. Really in Tokyo I would focus on things it is good at, i.e. mostly on modern ctiy stuff. Also, each of these places is in a different part of Tokyo, and you will spend considerable time moving from one place to another. In one day, I'd try to cover one or at most two parts of Tokyo and do a lot of walking.
5 DAY TRIP TO KAMAKURA
6 MT FUJI, KAWAGUCHIKO, CHUREITO PAGODA
All good day trips from Tokyo, but personally I'd rather have two days in Nikko (which has better nature and very good shrines/temples). I'd most likely sacrifice the Tsukuba and Kawaguchiko day trips for it.
7 SAIKO, TRAVEL TO MATSUMOTO LATE AFTERNOON VIA KOFU AND SPEND NIGHT IN KOFU
8 SHOSENKYO GORGE IN KOFU AND SPEND NIGHT IN MATSUMOTO
I don't understand this part. Why do you go twice (on day 7 and 8) to Matsumoto?
9 MATSUMOTO CASTLE & JOYAMA PARK, SHIRAHONE ONSEN IN THE AFTERNOON
10 SHIRAHONE ONSEN BACK MATSUMOTO VIA SHUTTLE BUS, MATSUMOTO TO TOYAMA VIA NAGANO, MONKEY PARK IF TIME ALLOWS, SPEND THE NIGHT IN TOYAMA
If you can skip the monkey park, you could travel Shirahone Onsen-(Kamikochi)-Takayama-Toyama instead
11 KUROBE GORGE (EXPECT 11 NOV)
Could probably do something else on this day, or at least go to Kanazawa.
12 TOYAMA TO NAGOYA VIA KANAZAWA, KENROKUEN IN KANAZAWA, HIMI COASTLINE IN THE MORNING
13 DAY TRIP TO KORANKEI VALLEY, OBARA IF TIME ALLOWS
Places like these are fine if you have the time, but I'd rather have more time in Kyoto
17 HALF DAY NARA, HALF DAY UJI
Nara has enough to keep you busy for a day or more. I would not try to split time between Nara and Uji as you'd lose around 1 hour or so on transport.
18 KYOTO
19 KYOTO TO SHINGU, KUMANO KODO
20 DAY TRIP TO NACHI FALL, HONGU
21 SHINGU TO ISE SHIMA, VISIT ISE GRAND SHRINE
22 TOBA, AMAHUT, ISE BACK TO OSAKA LATE AFTERNOON
You could investigate whether it would make sense to travel this in reverse direction after Nagoya/Kiso Valley.
23 DAY TRIP TO KOYA SAN
The main point in Koyasan is the temple stay/early morning service. IMO you lose a lot if you go just for a day trip. The temples themselves are not all that special compared to what you can see in Nara or Kyoto.