Shinjuku is probably one of the busiest areas, good for shopping and dining, and is usually where I stay since it's very central.
Here's a link to Japan Guide's page on Shinjuku. Shinjuku station is the busiest station in Tokyo, so it meets the bustling life criteria, Golden Gai and Omoide Yokocho have tons of small eateries if you want the street food ...
Some, if not most, stores are open.
Note that I haven't visited Harajuku on New Year's, so my information is second-hand, and most of my references are in Japanese.
Looking through Yahoo Q&As (like this and this), I'm getting mixed signals: some says a lot of shops are open, and others say most are closed (also note that some of the ...