The catapult has several uses:
When lit, it can be used to aim and shoot at the Tiki Guards. Use aiming to hit the guard that faces the catapult. Also use the flipper to hit the guards again before they sink back into the table.
If it is lit while the Tiki Guards are raised to full height in defense of Big Kong, you may be able to strike two guards at once by aiming in between them. While it takes three hits of a tiki guard to lower it one step, it only takes one hit from the catapult to lower a guard by one step. The guard also seem to lower one step if you can use the upper flipper to hit a guard again just after the catapult was used, like a rebound in basketball.