I think the first thing I would try would be replacing the coil assembly on top of the engine. Its a distributor less ignition system, and that coil assembly is not hard to replace. They are pricey though. One from a junkyard would be much cheaper to try. They dont fail very often, but then neither do fuses for the engine ignition.
Also, you should check to see if any 'codes' are stored in the computer. How to do this, and where to find a list of the codes, is explained in the answer to the 5th question in the FAQ Posting. Its at the top of this same subforum, for 1st generation Neons. Even though the Check Engine Light is not on, doesnt mean there is not a trouble code stored.