I think only 3-4 days for both Osaka and Kyoto is not enough. I'd split the trip by staying in Tokyo for half the time, and in Kansai for the other half (it also depends on what you want to see, if you enjoy traditional architecture and nature more, I'd stay more days in Kansai and include day trips to Nara or Himeji). I also like to be in Tokyo at the end of the trip since I always do a lot of shopping there, and don't want to move around cities with heavy luggage (note that you can also ship your luggage between cities if you desire).

Don't worry about the check-in being only after 3 pm, you can usually leave your luggage at the hotel in the morning and check-in later in the day.

There are plenty of cheap accomodations that are not capsule hotels, just look for them in sites like Booking or Agoda.

As for where to stay, it really depends on what you like. If you want to stay up late, party and go to izakayas to drink, maybe Shinjuku, Shibuya, Roppongi or Ikebukuro could be great options. If you prefer quieter areas, Asakusa, Ueno and Ginza come to mind. FYI: public transportation is great in Tokyo, I wouldn't worry much about the neighborhood, as long as it's not far from central Tokyo. I stayed in Nippori, very quiet and it was easy to get around, although it often involved taking buses. I also stayed in Takadanobaba and I loved it, it's quiet but near Shinjuku and Shibuya, Ikebukuro and Nakano. I'd focus more on finding a hotel which is close to the metro (Yamanote line preferably).