I currently have a '98 Neon with 135,000 miles (approx. 217,215 km) that I have had very little trouble with. You will of course have to keep up with basic maintenance (as you would with any car), but if you treat the car good it will treat you good. I have heard of and seen Neons over 250,000 miles (approx. 402,250 km) that are still running great.
The big issues to look for when purchasing a Neon: Head Gasket - Has the head gasket been replaced? And if so was it replaced with a newer MLS type gasket? (This is mainly just a problem on 95-97 & some early 98 models)
Motor Mounts - Automatics have three motor mounts, Manuals have four. Two are easily checked simply by looking/touching, inspect the passenger side motor mount located near the drivebelts. Try to fit your fingers actually thru the motor mount... there should be some small openings in the mount, but not even a single finger should be able to completely make it's way thru the entire motor mount. The front motor mount is underneath the front bumper in the center of the car... attempt the same test on it as well.
The only other thing I would mention is as Neon's become older and older they do tend to vibrate and make a little more interior noise than in their younger year

. No biggie tho, its not that bad.
L8R,