I recently moved quite close to the Daikanyama neighborhood (between Ebisu, Shibuya and Nakameguro) and I am enjoying it very much. There are so many cafes, boutiques and quiet streets to wander. That said, it is not the neighborhood to visit with the intent to do much more beyond eat, drink coffee or browse the shops. It feels like a great neighborhood for those who LIVE in Tokyo, but for those who wish to sightsee, I think it would become boring quite quickly.
Actually, I don't think there's much to see in Shinjuku Station or Shibuya Station, maybe just wander around big shops like Bicqlo to see the latest, well, fridges and TV's, and boutiques with fashionable clothes and shoes. In Asakusa I know only Sensoji and tried to look for some other interesting places, but there are only interesting old shops with miso etc, and a cat cafe.
There is a bunch of great places in Asakusa! A place I enjoyed visiting so much was the drum museum Taikokan. One can even try playing the majority of their exhibits which was quite unexpected for me.
If you are interested in temples and shrines, then Asakusa is definitely the right places to go. One of my favourite shrines is Imado Jinja. To my mind, it looks really cute because there are cats depicted everywhere.
For Shibuya, I quite like the back street of Nonbei Yokochou. As the surrounding area has developed, this has a feel that probably hasn't changed since the 60s/70s Showa era. Right next to Shibuya station too, worth even just walking up and down for a look (which would take all of 5 minutes to do so :)
Within Shinjuku, many samurai once lived in Kagurazaka and so the area is full of history. There are some hidden cobblestone alleys and a small geisha community. It is also know as Tokyo's 'French Quarter' and has many delicious French restaurants. Kagurazaka is easily accessible from Iidabashi or Kagurazaka Stations.