A guy I used to work with had a 99 Neon RT. It was junk really, nothing but problems, he had 2 headgaskets in about 80K miles and all kinds of small things like door locks sticking and window switches going out. Everything in that POS rattled too. He hated it.
Another guy had an 01 Neon, regular one. It blew a headgasket at about 11K miles, and the second at about 30K and was traded with coolant in the oil. No other problems on that one, but that would have been enough for me

There is nothing like paying too much for problems. I would not consider a neon, no way.
I have rented them around the country several times and still cannot get over how cheap and tinny they are. They are also gutless unless you get one of the RT's with a 5 speed, then they do OK. I rented one in Phoenix and went up to Prescott in it, there were stretches where I had it pinned for 5 or 6 miles without letting up at all, in and out of second gear trying to maintain about 65-70. It puked out about half of the coolant on that trip up, but a couple big bottles of Evian from the gas station and I was off again. The transmission fluid was also burnt bad, it was putrid smelling. Forget using the AC on a 5% grade too, you will get run over by traffic. I took it back and made them give me another one, it was just as bad so I took it back and got a Pontiac Sunfire. The Sunfire was twice the car that the Neon was, and got better mileage!! I have driven at least 8 different Neons and they were ALL junk, you could not give me one.
Last story, I backed into the 99 Neon at work and had the hitch solid against the intake manifold before I even felt it. I was idling on flat ground with a stock auto, about 5 feet away when I started backing and it went that deep!!! All in all it cost my insurance company about $2800 to repair the Neon, the paint on my hitch was not even scratched. Poor guy had just gotten it out of the shop after getting creamed by a big work van!!! I was feeling bad until he was mad that I did not total it, then it was kinda funny.
OK so it wasn't the last story

, but both guys were severely upside down on the Neons the day they bought them. The 01 sold for about $14K, it was as loaded as you can get one, with 11K miles on it trade value was about $6500 at the dealer where he bought it and all the others were somewhat lower. Lottery tickets would be a better investment IMO.
Hope this helps, I am not trying to be incredibly negative but the true nature of my experience with them is just that. I would hate to see anyone get the shaft in Neon form again.