Monitor works perfectly for 10 mins to 2 hours at a time, then screen compresses to about half size suddenly at the sides, but still touches top and bottom. I can get it to return to normal by tapping the large circuit board just underneath the back on its outer edge with a plastic ruler.

Sometimes it turns off completely and a have to turn off power and wait a few minutes before turning on power switch on again.

When trying to turn power on again, sometimes it comes on for a half second then shuts off.

Screen also shrinks momentarilly (in both vert and horiz) when maximising windows.

Lowering Brightness and Contrast does no seem to reduce the incidence of any of these symtoms.

I am only very slightly technical, so I would like to know what my options are. I would hate to throw away a 21' monitor if the problem is relatively easy to fix, or can be fixed for less than $150.

Thanks very much in advance for any suggestions.

I would not use it in this condition as it really will cause more damage as it will eventually kill the line output transistor which starts a chain reaction of destruction back to the power supply.
The problem you have will almost certainly be dry solder joints a common thing in these. Resolder any bad / cracked joints and that should fix it. As for the screen size changing, this is caused by failed capacitors on the power supply's secondaries. The one that fails most often is the capacitor on the voltage rail feeding the regulation feedback circuit to the power supply.