How can I test if the ecu computer needs to be replaced on a 1992 corolla? The car starts but only runs for a second or so.

You won't be able to 100% test the ECU yourself. Carry out all the checks and you will end up with a balance of probabilities.

The ECU can't output signals unless it is receiving proper inputs and has a good supply and ground, sometimes several of each.

If the car has an immobiliser it is something else to check and possibly worry about because if it isn't disarming it will seem as if the ECU is dead. Mainstream manufacturers were fitting a PATS system to some of their products by 1992.

You need a wiring diagram and some decent test gear to check the continuity of every wire that could be suspect, check every voltage, check every ground, every sensor - it is a long tedious process. Checking the inputs and outputs of the ECU with an oscilloscope helps.

From your description, if I had to guess, I would suggest the signal from the distributor or crank/camshaft sensor is being lost.

You can't test it at home, it requires special equip.