Uber gives a flat rate of $65-$125 between JFK and Manhattan depending on the type of car requested. I believe that cabs will take you there for $45-ish, but will expect a tip.
For Manhattan to EWR, Uber does $85-$150, probably less for cabs by the same proportion.
Neither of those services require (or accept) reservations in advance. There were also be plenty of people in the airport saying, “you need taxi?” to everyone who passes through. It is best to ignore them and go to the taxi stand.
I haven’t flown in and out of NY for some years but I doubt you’ll have problems getting a cab or limo at the airport without a reservation. Depending on conditions the wait may vary. There are often cars available virtually instantly. I remember waiting about an hour once long, long ago. Things may have changed since those days.
Also, I wouldn’t call any of those rates reasonable.
There are official NYC cab stands at JFK. Cabs are constantly available, any time of day or night. There’s a flat fee for the trip from JFK to Manhattan, it’s $52.50 plus any tolls if applicable. From JFK to Newark isn’t flat rate, and I would guess it will be pretty expensive. (more than $100? But I’m guessing). In addition to the travel charges, because its outside of New York City, you will be obligated to pay the cabbies tolls in both directions.
Separate from the official NYC taxis, there are also some car services located in the Terminals. They do accept walk-ups generally speaking. No idea of their usual fee for such a trip, but in my experience, car service and official taxis charge comparable fees for comparable trips.
Finally, from JFK airport you can take the AirTrain to Jamaica Station and get the LIRR into Penn Station quite easily and comparatively cheaply. From Penn, NJ Transit takes you to EWR.
Once you get past the baggage claim area, there is a ground transportation desk. There are phones that will directly connect you with various car services and door-to-door shuttle services. Also, there are red-jacketed people there to assist you with arranging ground transportation.
So far as I know, time of day has no impact on car service rates. There are various surcharges on taxis at different times (night, rush hour), but we’re talking $0.50-1.00–probably 2-3% of the total fare.
Note that there’s no fixed rate for taxi trips from the City (used in its inclusive sense here to include the Boroughs, and by extension, JFK in Queens) to Newark–you have to negotiate it with the driver. You can also almost certainly get from EWR to Manhattan, at least, much cheaper by taking an Amtrak or NJ transit train. No idea how frequently they run, though.
ETA2: Also, keep in mind that car services expect tips as well. The one referenced in post #2 just includes it in the flat rate.
NJ Transit’s Newark Airport Service is around every 10 minutes, sometimes every 20 minutes; there’s no service between 2am and 5 am. Trains stopping at Newark airport are marked EWR on the “big board” in the Amtrak waiting area, including NJ Transit trains.