+1 to rrt above.

Any extra cash you might save using one of the more convoluted methods would be offset by the sheer hassle. You would be far better off just paying for your N'Ex ticket at the same time you activate your Rail Pass at the JR office at NRT. You will be there anyway and you can use a credit card for that purchase if you want to avoid cash. I suggest having some extra cash in pocket when you leave NRT but you shouldn't need much if you intend to use Suica for most purchases. Considering the disparate destinations in your itinerary, you are almost certain to run across someplace the card won't work.

And I would strongly suggest avoiding any route to the City that would take you onto the Yamanote Line if you have much luggage.