Sounds like it is missfiring. Have the compression checked, fuel pressure checked and spark tested. I bet it needs a good old fashion complete tune up
Integrity Auto Care
September, 11, 2012 AT 3:14 PM
I am working on an '03 Neon right now with the same problem. The compression on cylinder #1 is low; not sure why yet. I would check compression (or have a mechanic do it) before you replace any tune up parts. There are quick and simple ways to determine which cylinder is misfiring without replacing any parts and to determin whether it is the spark plug, plug wire, coil, fuel injector, or something internal (low compression).
October, 13, 2012 AT 1:56 AM
PUT NEW PLUGS IN, IF NO IMPROVEMENT IN PERFORMANCE, AND DEPENDING ON MILES TIMING BELT MAY BE STRETCHED AND READY TO FAIL. THE KILLER OF NEONS