This is actually a simple procedure on the Neon. Here are the detailed instruction, taken from a website on Neon troubleshooting and repair:
"This is pretty simple and can be done by anyone who has some kind of gauge problem and a soldering iron.
Remove your gauges/cluster: Pop the upper part of your dash off. There are no screws, it just pops straight up and out. Put it in your back seat. The cluster had 4 screws holding it in. The hardest screw to reach is the bottom left screw. I used an offset screwdriver to reach it. Once those are out, the cluster just pops out. You don’t have to worry about unhooking anything.
Flip your cluster over. Remove the paper protector from the cluster. The screws are #15 Torx head screws. Once the paper is removed, remove ALL of the remaining screws. I believe there are 9 of them, also #15 Torx head. You don’t have to remove the bulbs. Disconnect the 4-pin ribbon cable. Slowly and carefully work the circuit board off of the pins. Try not to crack the circuit board. Once you break the board, this little job is done and you’ll have to find a new cluster.
Once you have the circuit board loose, flip it over and heat up your soldering iron. The points you need to reflow are the 20 joints arranged in four rows of five.
Set your hot soldering iron onto the very top of the soldering joints. There is probably a wire nib sticking out of the solder. Use that. Watch the solder, it will appear to change color. That’s all. It won’t move or anything, just change color. Don’t add any solder to it. Make sure it changes color all the way down. Set the circuit board off to the side for a few minutes to allow it to cool, the reassemble it the opposite of disassembly.
You’ve just successfully reflowed the cold solder joints in your cluster, and you shouldn’t have anymore jumping gauges. If your gauges continue to jump or flicker, then you’ve A) found a way to screw this up or B) have a cracked circuit board and you need to replace the entire cluster."