I have a 16 year old side by side unit that I suspect has a faulty thermostat. I am only a novice and home owner. The unit does not come on and shut off like it should, but I noticed that if I tap on the panel where the temp. controls are located, the unit does come on and runs fine, but it may run for several hours before it shuts off. The next day, I have to again tap on this panel to make the unit run. Once I force it on, it cools fine, makes ice, etc. but there is definitely some problem. Before I have a service man come out, I would like to able to explain to them the problem. Does this make any sense? Any suggestions?

Perhaps try gently turning the control up/down a bit (not to off).
If it fires up then the control thermostat is the more likely cause.

I am somewhat mechanical, but is replacing this cold control something that I can do, or does it require a serviceman?

I tried turning the refrigerator temp. control knob to "coldest" and then back to #2, back and forth about 5 times but the unit did not start. I then tapped the panel and the unit did start, and ran for about three hours. I noticed that later in the day, the unit was running without having to tap the panel, or force it on.

Yes it could be a bad connection.

Or since the temperature control did not start the unit up, it could be a bad defrost timer.