Email from my husband to my brother who's coming to Tokyo next week from New York:
The [Narita express train from the airport] splits at Tokyo, with half of the cars bound for Yokohama (south) and half for Shinjuku (north), so you can take the N'EX 30 (or the 32 or the 34) and get off at Shibuya or Shinagawa. Just go to the counter and tell them Shibuya and they'll give you a seat in a car going to Shibuya - just make sure you sit the designated one (it'll say on the ticket and the clerk will helpfully / painstakingly point it out). You can pay with a credit card.
But really either station is fine - it's not that far to the Westin from Shinagawa, although it will be more than 1,000 yen. It could be more from Shibuya as well, so you should take out money at the airport (if you forget you can use a credit card, by the way). Just say "Card-o OK?" and look apologetic. If they say bakka gaijin that means that they are welcoming you to their country with open arms and hoping to see you in their cab again soon.
But I do recommend you at least consider the bus. I understand and agree with you about trains versus buses. But you're not getting on a crusty Gray Line bus from JFK to the Port Authority, with some obese dude noisily eating a bag of Doritos in the corner. You'll be boarding a gleaming Friendly Limousine Bus where a 75-pound girl will smilingly lug your bag from curb to cargo hold, and you'll be taken directly to your hotel. No muss, no fuss.