I had a car that had a leaky injector that had problems similar to what you are describing.

I never actually fixed it, I just learned to turn the key to run for about 3 seconds and back to off 2-3 times to pressure the fuel system back up.

Someone mentioned leaving the key on for 30 seconds to build pressure - I don't know about your particular model, but mine would only run the fuel pump priming cycle for about 2 seconds, this is why I had to flip the key back and forth.

Not a fix but maybe this will point you in the right direction.

Thanks,
Ryan

1984 Mitsubishi Montero, 2.6l - 5 speed