We also recently stayed in Shinjuku and we simply caught the train.
We found it very easy, although it was our third time in Japan. It does make for a long day if you're at the park an hour early to get in line and stay there until after closing looking at the shops.
Most basic way is to take the green JR Yamanote line to Tokyo station and follow the pink signs (or people with lots of Disney characters on their bags) to the Keiyo line for Maihama. Make sure you get your tickets ahead of time at a Disney Store too. Disneytravelbabble.com has a good summary of her trip.
The train system is easy, but the transfer at Tokyo station to the Keiyo Line is a long walk. There are not many breakfast options in the park so grab some grub before you go. Dinner is usually heavily reserved, but you can get a nice meal for lunch at the SSS Columbia (in American Waterfront part of the park) as early as 11:30. If you get a window seat, you can view the park and may catch a sight of Mickey if they are having a show performance during your meal. Get fast passes for Toy Story Mania right away and then head towards the back of the park.
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