I'm not sure that my car even has OBDI. Isn't it too old?
Nope, you have OBDI. What you're too old for is OBDII (96+)

And doesn't the check engine light have to stay on to pick up the code?
Not necessarily. Some codes are stored, some are not. Its defiantly worth checking. RL7 linked you to the procedure, just keep in mind you MUST use either an analog meter as shown, or a low voltage test light.

I know my dad has one but I think it is OBDII only but I could be wrong.
Chances are its OBDII and won't work for you.

This is a SOHC with a cap and rotor. right? How old is the tune-up? (Plugs, wires, cap, & rotor)