The Park Hyat is giving you a deal.

Again the definition of a limo is in order here

Limosuine from the north american perspective. There are 3 types

Lincoln Town Car (classy sedan chauferred)
Regular Limo (your standard sized limo, chauferred)
Stretch Limo (much longer limo chaufferred)

In Tokyo there is only the first, you will more than likely not get a 'real' limo as you may expect in say New York or somthing.

The closest I have seen to a limo is what i would consider a "stretched sedan", it was longer than a normal sedan vehicle, but not as long as a 'regular' limo. It looked like a midget limo.

I have been in one of those, for a 22 min ride from central tokyo to another area in central tokyo and then back again, was near 600 dollars.

If park hyatt does dispatch a vehicle to you, expect nothing more than somthing only slightly larger/nicer than a taxi.

Honestly if you want to go first class into Tokyo, get a F class ticket from NRT to Haneda

Otherwise the Narita Express green carriages arnt bad.