Pump surging and not priming itself

Fuel filter may be plugged. Pumping up a fuel tank will cause damage to the fuel pump diaphragm , the gas tank itself. Priming the carb is done by pouring gasoline in the vent hole of the float chamber.

There is no need to prime. But you do need to use the black spray nozzle to get soap to siphon out.
if that doesn't solve the problem let me know

You have to use a low pressure nozzle ( usually black with very large spray opening) With the low pressure nozzle siphons your detergent in with the spray water . If you are using a high pressure nozzle the soap will not prime.

Airless paint sprayers should not turn off while in prime mode. As long as the prime valve is open the motor can pump paint but the system will not build any pressure. Once the prime valve is closed the pump should pump paint into the hose until it gets to the pressure level required to turn the motor off. If the pump does not turn off with the prime valve closed, it could be that the spring in the valve is weak and won't hold the valve closed under pressure. If the hose is up to pressure and the motor just keeps trying to pump paint, you may have a bad pressure sensor or it is disconnected from the circuit board.

All airless sprayers are on-demand pumps. The two settings are prime and spray. When it is set to prime, the pump will draw liquid from the bucket below and pump it out the side tube and will never stop until you switch it back to spray. When set to spray, the pump will run until the pre-set pressure is reached, then stop. As soon as you start spraying, the pump will run until a few seconds after you stop spraying. Tip size and pressure setting will affect how long after you stop spraying the pump will run. A large tip and high pressure will cause the pump to run longer after you stop spraying.

It sounds like the fuel pump is not working. Perhaps the install was not completed correctly. Things to check, fuel pump relay, wiring, voltage to the pump.
The priming happens as soon as you put the key in and turn it to "acc" (you may hear a humming sound). Then once you turn it to "start" the injectors and spark plugs start firing.
Another possibility is that the fuel filter was installed backwards.

Turn the key on for 5 sec. Turn the key off for 20 sec. Repeat 3 times

You have a defective pressure switch.But replacing the switch. Do the following; Open the pressure switch cover, and find the two adjusting nuts. One is for the cut-off pressure (40psi) and the other is for the cut-in pressure (20 psi). Adjust the the nut with a longer stem counter clockwise (loose) with the pump running. Take care not to touch the power line attach
to it. Continue the adjustment until the pump stops,
at 40 psi. Then, observe - Pump must stop once the
pressure is attained. Then, open the valve until it
start (20psi).. Then you can adjust the other nut to
cut in at 20 psi.
To prime the pump; open and fill the priming port full
with water.and replace the priming port plug. Set
the water flow control valve at full close position.
Switch the power on to run the pump and slowly
open the control valve to appropriate setting such
that the pump doesn't lost the suction. Send your e-
mail for further comment. God bless u

if it has a prime bulb on it the rubber may have deteriorated and is failing to prime?Just my first "guess" hope it helps (replacement bulbs can be found easily)