Actually I kind of enjoyed walking into Mom & Pop restaurants and picking stuff from the menu with little or no clue what it would be, and I’m rather picky with food normally.
I second the reccomendation of getting a JR Pass, it’s just wonderful to get on most trains, including the Shinkansen, at any place and time.
A few more general tips:
-Get an umbrella. You’ll need it. I’m betting your tour guide won’t let a little rain interfere with the itinerary.
-Bring multiple handkerchiefs. You’ll need them.
-Japanese cities are safe, but very easy to get lost in as you wander away from the station. Only major streets are named and addresses are kind of notional. The ancient capital cities of Kyoto and Nara were laid out in block grid patterns, but Tokyo and Osaka are certainly not. You can wander down an interesting looking alley and have no idea where you end up.
Even Japanese people will often meet at a central location before going off to find the restaurant or whatever they are going to.
As to the specific free days in Kyoto, Osaka, and Tokyo. I’ll run through some possible suggestions, but a lot depends on your preferences. Both Osaka and Kyoto have distinctive food cultures that you should make an effort to sample.
Osaka: I didn’t really spend too much time in Osaka beyond wha’s on your itinerary, although I did go there a few times for concerts. I went to the aquarium, but I really don’t think that would be worth your time. Other than Dotonbori at night, or a Hanshin Tigers baseball game, I don’t have much to recommend.
Kyoto: In Central Kyoto, I’d suggest Nishiki Market, which has a lot of traditional Japanese food sellers that have samples if you’re feeling adventurous. You can stroll from there to the traditional areas of Pontocho and Shirakawa, where you might catch sight of a geisha or maiko if your timing is right.
For longer walks, you can either do the east side (Higashiyama) or the west side (Arashiyama). The Higashiyama walk is centered along the Philosopher’s Path (Testugaku no Michi), which is a pedestrian path along both sides of a tree-lined canal. It runs roughly from the Nanzenji Temple complex to Kinkakuji (Temple of the Silver Pavilion), with several other temples, shops and small museums on the way. You can extend the walk to the south from Nanzenjio the area of Khiyomizudera, perhaps the most memorable of the temples not on your tour. The Arashiyama walk goes across historic Togetsukyo Bridge and then to Tenryuji Temple and Arashyama Bamboo Forest. There are more options depending which way you want to go and what you want to see.
Tokyo: Pick a neighborhood or a string of neighborhoods, and have a good walking map or street map. A lot depends on which Tokyo you want to see. The modern, youth-oriented entertainment districts are Shibuya and Shinjuku, plus Haraujuku for outre youth fashion and Shimo-kitazawa for vintage goods, indie music clubs and so on. Odaiba is the ultra-modern reclaimed island out in Tokyo Bay that is mostly an entertainment district and noted for over-the-top post-modern architecture.
Akihabara/Kanda/Jimbocho: Akihabara is the well-known discount electronics district which is also a main hub for gaming and anime otakus. Kanda and Jimobcho have a long array of used bookstores and one section of Kanda has some of the oldest restaurants in Tokyo.
Yanaka/Nezu/Sendagi: Perhaps the neighborhoods where you can see the most of old Tokyo, especially if you stick to the back streets.
Tsukuji Fish Market: Even if you can’t get there at the crack of dawn to see the busy wholesale fish market, the Outer Market is still worth seeing. From there you can walk to either Ginza, or see what’s left of the old island neighborhood of Tsukishima, or visit Hamarikyu Gardens (from where you can catch a riverboat to Asakusa.
Askausa/Kappabashi: Your are already going to Asakusa to go to Sensoji Temple, but will probably take the main path from to the temple, which is mostly lined with booths selling tourist tat. The Asakusa neighborhood was the heart of the Edo period Yoshiwara pleasure quarter and then the heart of Tokyo’s shitamachi (low city), and you can find some of that wandering around the back areas. Kappabashi-doori is the heart of Tokyo’s restaurant and kitchen supply business, where you can buy anything from plastic display sushi to high-end cutlery.
Ryogoku/Asakusabashi: Ryogoku is sumo town. The sumo stadium is there, as are a lot of sumo stables, where the wrestlers live and train (although they’re not open to the general public). If you want to try eating like a sumo, there are chanko-nabe restaurants in the area. The very impressive Edo-Tokyo Museum, which is an indoor reconstruction of an Edo neighborhood, is next door. There are parks and gardens around with some other interesting-sounding museums (Sword Museum, Hokusai Museum, Kanto Earthquake Museum). Across the river is Asakusabashi, home of traditional Japanese doll shops, fabric crafting supplies, and handicraft studios.
I loved Tokyo. The crowds were part of it’s charm, not unlike NYC. It’s a great place to people watch & shop & the Japanese were very kind & helpful. There is a huge fish market there (world’s largest?) called Tsukiji, which is incredible in that there are dead fish everywhere, yet no odor of fish. People are running around cleaning constantly & there is ice everywhere. It’s a pretty cool operation to watch. Enjoy your trip!
Finally (whew), some possible short trips that should take less than a full day.
Uji: A few minutes south of Kyoto. Uji is like a smaller Kyoto but without the crowds. The standout World Cultural Heritage sites are Byodo-in Temple and Ujigami Shrine (Japan’s oldest shrine building). It’s also home to the Tale of Genji Museum (the last part of the book took place in Uji) and a center of tea production, so there are tea shops and the like. Like I said, it’s a lot quieter than Kyoto.
Kobe: A little west of Osaka. I used to go there for annual work conferences and to visit friends. For its size, it’s both a manageable (narrowly situated between the mountains and the sea) and cosmopolitan (one of the first Japanese ports opened to the West.) city. The shopping/nightlife/entertainment districts are down by the water, but the more interesting stuff is up by the mountains. There are ropeways and a cablecar up Mt. Rokko and Mt Maya where you can get really good views and there are a few attractions you can visit by ropeway, such as the Nunoike Herb Gardens.
You can also visit:
-Nada Sake Brewing Distict
-Kitanocho: The old foreign district. A lot of the former homes and consulates now host small museums, and funky shops, boutiques, and restaurants.
-Chinatown:
If you want an actual Japanese mountain hot springs experience, Arima Onsen just outside Kobe is your best bet during your free days. You can get there by taking the Mt. Rokko cablecar and then take a ropeway down the other side of the mountain to Arima Onsen.
Hope all this rambling helps.
I’ll second the recommendation for Nunobiki Falls/herb garden. The base station for the cable car is just off the west end of Shin-Kobe Station. Take the cable car (“ropeway”) to the top, and enjoy a long winding walk down through herb gardens, past the waterfall, and back to Shin-Kobe Station.
Given the recent typhoon flooding/damage in the Kansai area, you’ll want to check on the status of some of these attractions (esp. Rokko/Maya) before going.
I’ll also add the Akashi Kaikyo bridge tour. This is the longest suspension bridge in the world; it is ridiculously big when you’re standing beneath it. Many of the tour guides are retired construction workers who actually built the bridge, so they’re pretty proud to show it off. The tour starts in a museum that documents the bridge’s construction history and technology. Then you are guided as a group out onto the maintenance walkway of the bridge, out to one of the two main towers (it’s a long walk on open grating a few hundred feet above the ocean), and then you take an elevator to the top of the tower for some grand views of the bridge, Kobe, Osaka Bay, and Awajishima. English-language audio players are available. Reservation and fee required, but (IMHO) totally worth it.
Curious as to what OP did on their free days in Japan, unless the trip was hindered by the typhoon hitting western Japan that cut off Kansai International Airport from the mainland after a cargo ship was rammed in the bridge.