You can't "prime-up" a starter. Its an electric motor. What's probably happening is you're letting the fuel pump run for a second & getting fuel flowing in the rail. If your problem is plugs/wires/cap/rotor or anything like that, then running the fuel pump won't make any difference.
I haven't messed with that yet on the Integra, but on my former '89 Saab it worked like this...
1) Fuel pump had a check valve, to prevent gas from flowing back into the tank when you shut off. That might be leaking.
2) Fuel pressure regulator isn't closing completely, so fuel-rail pressure is leaking down thru the FPR.