I'd pick Shinjuku as well. I lived there for 3 months last year and really enjoyed it.
Great food, great shopping, vivid nightlife (if you choose to seek it out, it's not 'in your face') and *amazingly* well connected to public transit. Shinjuku Station is really the heart of Tokyo's train system. Mind you, it is INSANELY busy. Imagine a crowded train station, like the busiest one you've ever seen. Now multiply that by 20.
I wrote about Shinjuku on my blog while I was still living there. I think my description then is a lot more detailed: http://www.thatsmycoffee.com/trips/life-in-japan/
I stayad at Ikebukuro last January and I loved the neighborhood. My hotel was The Hotel Metropolitan Tokyo and they have have bus ( limousine bus) to/from Narita or Haneda Airport. It iwas a good choice. Good Luck
I'm also recommend Shinjuku would be better for you. However, the noth of Shinjuku ,Kabukicho , is a little bit dangerous area. So, you should book the hotel nearby the west of Shinjuku. For example, Washington Hotel
Personally, I would avoid staying in Shibuya or Shinjuku. Both are amazing areas to explore for myriad of reasons but if you are looking for a real Tokyo neighborhood experience there are better options. I live in the Koenji area and I would highly suggest it. There are tons of cool shops, restaurants, and amazing bars all within walking distance of the station. It is kinda hipster...kinda punk rock... but with families and regular people also living in the area. It can seem a bit quiet at day but thats just because most of the shops don't open until the afternoon. As far as safety goes, you are in Japan! When I travel home to Chicago, I forget how much I've allowed myself to let my guard down in public as a result of living in Tokyo. There really isn't anywhere in Tokyo you could stay that wouldn't be safe for you. Other places to research: Shimokitazawa and Nakameguro. Have a great trip!
Right around the Jinbocho Station, there's a place called Sakura Hotel. It's very friendly and reasonably priced for a Japanese hotel. The proximity to the subway is great, and it's a very nice place to walk. There are plenty of nearby places to eat, and it even has a pretty decent espresso bar in the lobby. You'll be able to stroll pretty easily to some gardens and parks. My wife even got a few hourlong runs in from there.
I think Shinjuku would be your best bet. It's a lot of fun and very convenient. My favorite place to stay there is the Granbell Hotel Shinjuku which is very reasonable. I think the cherry blossoms will be in bloom while you're there so make sure you get to the Shinjuku Gyoen National Garden. The cherry trees there are beyond belief.
I also suggest that you to stay in Shinjuku, particularly in West Shinjuku.
Shinjuku is pretty accessible whether you are traveling within Tokyo or for day trips to see Mt. Fuji.
To get a feel and see the difference between popular neighbourhoods in Tokyo, you may want to check out this Where to Stay in Tokyo blog.