For a first time visit, I'd chose Asakusa of the two neighborhoods your selected. It's convenient to both airports via train (Narita is a bit more direct than Haneda and you can do with 0 transfers), is close to Tokyo Metro, and while definitely quieter than Shinjuku or Shibuya, still has something going on at all hours of the day if you wander deep enough in. Prices are also quite reasonable, and the proximity to the shrine and the tourist stuff that accompanies it makes for a really nice daytime environment.

I stayed at the Hotel Hokke Club Asakusa, they're a Japanese chain with a bunch of locations. Very clean and comfortable room, though Tokyo sized and less than a 5 minute walk to transport, 7/11s, restaurants and other things.