Sounds like the pump (really it is the one way check valve within pump) is getting ready to die. The check valve keep the fuel rail pressurized after shut the engine off until you start it next time. Stick a fuel pressure gauge on the rail and see what it is while you are cranking the engine over and experiencing what i call the dreaded carbureted start. It could also be a leaky injector leaking off pressure too. But this is the classic symptom of a pump getting ready to die.