Have you replaced the fuel filter? It sounds like the fuel pressure is bleeding off at shutdown. Can be a fuel pressure regulator problem.
There is no serviceable fuel filter on the 1998 Neon. It is inline with the fuel pump. The tank will have to be dropped and fuel pump replaced.
If the check engine light is or has illuminated, check for codes.
If the engine cranks & cranks & cranks before it starts even though it's warmed up already.
My car starts but it shuts off it says ck engine
You need a battery. There's a very small chance it could be something else, but from the symptoms you describe it is more than likely a bad battery.
What year? What is it doing? A 12 valve has a fuel shut off solenoid that commonly malfunctions, the engine cranks but never starts.
it could possibly be a bad coil, crank shaft sensor, or cam shaft sensor.
The fuse box under the hood. It is the #9 relay and when it starts to go bad there is a delay before engine cranks over.
Open a bit more the screw that regulates fuel input into the carburator.
crankshaft
I had a simalar problem on my 99 e320. It turned out to be a crankshaft position sensor. by the way, the engine cranks automatically until it starts (or doesn't) even if you let go of the key and let it return to the run position. other possibilitys include the ignition key
spark plugs are irridum and should be changed at 90-100k, trans fluid should be changed every 30k, coolant should be changed every 30k, srep belt should be changed when it starts cracking or starts to squeek
both keys have chip for security , 1 is bad