I took the Narita Express from the airport to Shibuya station and it couldn’t have been easier. Even if you don’t use a wheelchair, this is probably the most convenient way to get into the city from the airport.
There are very few stops and this train can get you to any of the major stations in the city (Tokyo, Shibuya, Shinjuku). The Narita Express is on the JR line of the Japanese trains. If you are arriving to Tokyo Narita, the process is simple — just tell them you need the Narita Express or follow the signs.
The ticket desk is just before the electronic gates to enter the train. If you need a wheelchair space you’ll have to stop there to make sure you get a wheelchair spot on the train. It seemed like there were two in the car that I rode in. The train runs often enough that you won’t have to wait long.
I normally like to book things online and deal with it on my own, but in this case I was handled by the train staff in a very deliberate and accommodating fashion. I never felt like I was going to be forgotten and the same guy who helped me with my ticket ended up leading me down to the platform and then onto the train with the ramp.
You can buy a round trip for 4,000 yen if show your visitor passport. Just make sure to hold onto the return ticket pass.

Returning was easy too but a little more confusing, probably because Shibuya station was in repair in anticipation of the Olympics. Nevertheless if you get as far as the ticket gate and don’t have a ticket yet, stop at the counter there and just ask for help.
The train was not crowded on the Sunday afternoon when I arrived or the Sunday afternoon when I departed. It was comfortable and quiet.
Here’s more info about the Narita Express and other airport transit options.
