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Some recs I passed onto a friend 2 years ago when I went, hope this helps:
Tokyo: Edo-Tokyo museum (very well done museum focusing just on Tokyo's history itself, free English tours, digestible 2hr focused history on the city from the Heian period through post-modern times), Akihabara electrical district, Shinjuku district at night, Shibuya district to see the giant intersection/crossing, try to go to a local baseball game and get wrapped up in the energy, overnight trip to surrounding area with onsens (spas)
Kyoto: Philosopher's path, Fushimi Inari Taisha (2hr roundtirip mountain walk with 1,000 red torii shrines), Tokugawa castle (if you're a history buff, most of the original mid-1800s shogunate style has been preserved), Golden Pavillion. There are a billion temples/sanctuaries in Kyoto so it's easy to get fatigued, I would recommend just focusing on 3 must sees unless you're a particular enthusiast. I didn't get to go to a Geisha show but that is something I want to do next time.
Osaka: This is a good city to live in but not necessarily fun to visit. If I could re-do my 10 days there, I would have skipped Osaka and spent more time in Tokyo and surrounding areas. We did 4 in Tokyo, 3 in Kyoto, 3 in Osaka. Since you have more time, check out the giant red torii in the ocean in the southern part of the country.
There is an endless amount of great food to eat. We did go to Jiro Sushi as well, and while it was an awesome experience and Jiro himself is delightfully quirky, if you're on a budget you can get better meal experiences (i.e. omakase but not in a subway station and with a chef who will talk/joke around with you) for less than half the price. That being said I'm not a picky eater. I also just liked the random sushi/snacks you could get out of their 7/11 style convenience stores.
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