Have someone pump the fuel primer bulb constantly, acting as a manual fuel pump. If that cures the problem, you either have a fuel/air leak between the primer bulb and the fuel pump, a fuel pump failed internally, or a blown piston that operates the fuel pump.
Does your warning horn sound off momentarily when you turn the key to the ON position?
The fuel primer solenoid is not at fault... impossible to have the engine run out of fuel UNLESS it's leaking and spraying fuel all over the powerhead.