Taxis FROM Manhattan to JFK run on the meter (the flat rate only applies for cabs TO Manhattan from JFK). Meter rate would depend on where you are and what the traffic is, but is generally in that range.
Chances are you'd get a newer, cleaner, fully stocked bar limo from one of the upscale companies, and they may be more reliable, though that's not a certainty. Sunday morning, though, is usually not a problem with any of them, barring weather.