Older pumps (or if you have too much debris in the tank?) may leak back into the tank, but should start quicker than that anyway. You could try mounting a fuel pressure gauge on the fuel rail to see if the pressure drops, and also try to prime it with a MAF strap to rule out other problems. (shorting the little two prong connector coming out of the MAF should make the fuel pump run as soon as you turn the key to ignition mark.)
Since it starts quick when you 'prime' the ignition, I actually doubt my suggestions will solve the problem..
Since it starts quick when you 'prime' the ignition, I actually doubt my suggestions will solve the problem..