For the sake of clarity - The train that I mentioned in my post is a JR train. It leaves JR Shinbashi Station at 6.06am and goes all the way to NRT with no changes required.

This is not the same as the route described by DL which is a subway and keisei route, which gets you to NRT earlier. (But the JR train gets there early enough).

As for getting to Shinbashi Station. It is easier and quicker to walk straight there from the Conrad than it is to take the Yurikamome line. Use the overhead walkways to avoid traffic and stairs.

If you have a lot of luggage you could take a cab to Shinbashi. It won't be much faster than walking.