If you're deadset on taking the train, choose the Century Hyatt and wake up early. The Narita Express leaves directly from Shinjuku Station at 7:07AM and 8:03 AM, otherwise, you have to pop over to Tokyo (about 15-20 minutes). The other hotels are a little more difficult, but none of them are especially "convenient" to Tokyo Station.

I'd actually recommend you consider the airport bus - http://www.limousinebus.co.jp/e/ . It's only 3000Y, and takes about 80-120 minutes depending on the hotel. However, they go directly from the hotels, so you don't have to run from the hotel to the train station and then hope you don't miss your connection.

No matter what, you're waking up early, and there's no way to avoid that.

BTW: No matter what, in all the cases, you'll have to do some walking (although I think the Yurikamome stops at the Intercontinental, you need to do the walking elsewhere on that one).