Craig, at Narita you can use the luggage carriers down the elevators and to the entrance of the N'EX train platforms where you roll your bags to your train car. I have never been to the N'EX gates in Shinjuku but there should be a straight shot to the taxi stands.....but nothing is a straight shot in Shinjuku station the ever evolving beast. This picture makes her look so calm.....
http://www.jreast.co.jp/e/stations/e866.html

Actually if you had no luggage, like lucyt thinks, taking the Chuo Express from Tokyo station would get you into Shinjuku quicker, but anyone who has taken the Chuo express knows, there are no reserved seats with extremely full train cars at all times of the day and doubly so during rush hours.

Aloha!

I've copied this in case "my man" gets lost again

Just for laughs, I printed out the Shinjuku Station map...

Looks easy enough, I know it isn't.

rhk, count on your man to get us lost and your woman to get us out of trouble as always......that's why I always take her along ;)

Aloha!

Craig and rhkkmk, Our hotel is in the direction of the east gate but you should use the taxi cue just outside the south gate when arriving on either track 5 or 6 on the N'EX.

Kathie, the Hyatt shuttle bus will drop you off at the west entrance which is a ways from the track 1-4 which your train to Omiya/Saitama should depart from. Allow yourself at least 1/2 hour to get from the shuttle bus to your gate. If your early no problem but walking through the station will take some time. Not because of the length but the confusion factor. Better a little early at your gate than late because the trains will leave precisely on time. You can set your watch by the train systems and most do in Japan.

Aloha!

I hope "my man" has back-up---wears a watch.

Marked this on my "map"...

Your man Smedley always wears a watch