Okay -- sounds like you'll do fine. So far your itenary is great. I would say Kamakura is the most important "just out of Tokyo" day. If you want to see "old neighborhoods" and whatnot, I'm afraid you'll find very little appealing to you in the city. The fact is, Japanese culture these days involves tearing down and building up more than it does preserving. Most everything was rebuilt after the war, and the mindset is "newer is better" around here. Kamakura though has HEAPS of these things, as does Kyoto. Looking at all your stops, I would say most of those places should serve as traditional spots, while for Tokyo you should concentrate on the modern edge. That really is how life is around here.
Some other stops:
Shinjuku: Great at night. This is the epitome of Tokyo, with blinding neon everywhere. Shinjuku Gyoen is the local park. It costs 200 yen to enter, but is well kept and is beautiful. Has a Japanese garden as well as a rose garden. Unlike most Japanese parks, they do not allow drinking, smoking, or sports.
Ginza: Shopping district. Similar to above, but a fraction of the size.
Shibuya: Young shopping district. This has "Scramble Intersection" which is famous for hundreds of people crossing at one time.
Harajuku: Even younger than Shibuya. This is where all the girls buy their ridiculous outfits. Personally it's a little too touristy, but oh, well. Meiji shrine is here which is definitely worth visiting. It is quite impressive. On Sunday afternoon, the Rockabillies will be dancing at the entrance of Yoyogi Park right right around the corner. Definitely worth seeing.
Akihabara: Electronics district. Back before the yen was crazy strong you could probably have bought loads of gear here cheaper than anywhere else!
Ueno: Little bit of everything. Very nice park.
Shimokitazawa: Hipster part of town.
Roppongi: Observation deck on Roppongi Hills. 2000 yen admittance (blech) and you will be very disappointed to know they do NOT allow tripods, as don't most places. You can get a panorama of the city here, including the Shinjuku skyline, but you will have to do it handheld.
Ikebukuro: Similar part of town to Shinjuku/Shibuya. Has "Sunshine 60" which has an indoor observation deck. Again, no tripods, but you can get some good shots through the glass.
There are many places I'm missing, I'm sure. Perhaps other people can chime in. I personally like to go to smaller parts of town I'm not familiar with and walk around. If you're not afraid to do that, there's probably 10-20 more spots you could do that, some of which are connected.
Depending on when you are here, there is likely to be some type of Festival going on -- or many. Possibly fireworks show. Be sure to check www.metroplis.co.jp
for information in English of what's going on!