My first impression on reading your itinerary is that it is a bit over-ambitious and some over-lap experiences.
The following are just my personal opinions:
Tokyo: Sky Tree and Metro Government Building. Chose just one. Both are tall lookouts. If you go to Sky Tree, then go to Kappabashi, the kitchenware street across the river on the west.
Meiji Shrine skip if you are going to Kyoto. Go to Heian Shrine there instead. In my opinion, prettier and has a beautiful stroll garden. There are no buildings to enter at shrines. They are basically a big courtyard with stalls around the periphery selling charms. If you go to Heian Shrine, then go across the street to the Handicraft Museum.
In Tokyo, go to Tokyo National Museum and/or Edo-Tokyo Museum. Take an architecture walking tour with Context Tours. Actually, this tour starts at Meiji Shrine. The guide will explain the blending of styles in the shrine. Tour Tsukiji Fish Market.
Or, if you're not big city people, skip Tokyo entirely. I'm the same. I had no desire to go to Shibuya crossing. I only went to Tokyo because my tour started there.
Kyoto: pare down the temples. You can stay overnight at Ninnaji. They have an English reservation website. Go to Kinkakuji before or after your stay, if it's on your must see list. Optional is to go to Ryoanji for the famous zen garden.
You can travel on to Arashiyama the next day for the bamboo forest, which will take only about 10 mins to see. The other option is to see Tenryuji, another zen garden here and skip Royanji. Or go to Daikokuji which has a beautiful garden with the oldest pond in Japan. This temple is not touristy and not crowded. Arashiyama is at least a half-day trip.
There is a website that tells you what temples are in reconstruction. I think Kiyomizudera is having major reconstruction. You might be able to read about current conditions on TripAdvisor. If you go to Kiyomizudera, continue southwest, and visit Kawai Kanjiro's House (a famous potter).
Other suggested activities besides visiting temples: Nishiki Market, take Taiko Drum lesson - really fun, Netsuke Museum, Nijo Castle with example of warrior garden. Take a day trip to Uji, known for their green tea. Byodoin Temple here is an example of a paradise garden. You can experience the tea ceremony here, but you can also do tea ceremony in Kyoto too. If you arrive on the right days, there are flea markets (with food stalls too) at some temples. There is a website for that.
I chose not to go to the more popular tourist-crowded temples, such as Kinkakuji and Kiyomizudera. It's not enjoyable for me.
I had a better time at Daitokuji. I chose a couple of the more famous small temples in the complex and got to enjoy the zen gardens pretty much to myself. It was really peaceful to walk the grounds.
There is a live webcam for views of Mt. Fuji. Most of the time it's cloudy. If you are going to Hakone, just in the hope of seeing Mt. Fuji, it may not be worth it. March may not be the best weather. But you could research that further. You could eliminate Hakone and go straight to Koyasan instead, then go to Kyoto. Check the weather in Koyasan. It's March. It may be pretty cold.
I feel there is a lot to see in Kyoto. One could easily spend 5-6 days there.
The people are very nice. Very friendly and polite. And no tipping anywhere! Clean public toilets at train stations, restaurants, etc.
At each of your hotel, pick up their business card in Japanese, show it to your taxi driver to get back.