I stayed at a capsule hotel just a block away from the station this summer, that was quite the experience. I was there for not even 3 full weeks but I think I'll feel nostalgic for Japan for the rest of my life, a strange and stunning country.
And still miss burnt miso ramen. Nowhere else in the world seems to do it.
(I have lived in or spent significant time in all of these cities you listed)
Now that I have lived there, all of that was very much confirmed.
What most people don't realize is everything there is around the capsules. Usually, when you enter the hotel, you start by going to the locker room. There you put in all the stuff you won't need for the night, that is, essentially everything. You may even get a yukata robe, a towel and some toiletries, so it can be literally everything, including your clothes.
Then you can take advantage of the amities and relax. You may find showers, hot baths, massage chairs, etc... Only when it is time to sleep you go to your capsule, which is essentially like a bunk bed with privacy, maybe with a bit of equipment like a TV, alarm clock, charging plug...
Now, I probably wouldn't stay in a capsule hotel for 3 weeks, but I could definitely spend 3 weeks backpacking, stopping in capsule hotels for the night. It is not really designed as a place to stay, more like a place to spend the night with no preparation. Also, while it is cheaper than regular hotels, it is not that cheap, and if you really want cheap, there are other options.
Often when I say that people tell me not to support the big hotelier chains and offer some independent options, which I might try next trip. I do think people might be a bit misguided though, independent or not they are still faceless hotel chains. Which one gets my $60 is hardly a matter of social responsibility.
If you are like me and are there for the urbanism, staying in a businesd hotel fits the atmosphere honestly. They're commoditized hotels, perfectly serviceable, pragmatic like much of urban life there.
It brings me to my other point. Capsule hotels are best when you are travelling alone. As a couple, they don't really make sense. In fact, some capsule hotels are for men only, or have men and women in separate areas. This is because they often feature public baths and there is some nudity involved, which is normal in Japan.
For a first timer, I suggest spending more time in Kyoto and Osaka and less time in Tokyo (but generally, you do Tokyo, Kyoto, Osaka on your first trip there and then diverge from that if you go back).
That and it was all so friendly, easy to get around, beautiful, etc. It’s a wonderful vacation for people that just like to walk around and explore.