How can I attach an air compressor to the pictured sprinkler system
That is a Vacuum Breaker which is in place to prevent water from siphoning back from the sprinkler system into the water supply. It closes under water pressure and opens to the atmosphere under vacuum.



You theoretically could spin the top off and using a rubber ball fitting, use an air hose to apply pressure, however; the inflow side of the circuit will have a manual or electric zone control valve preceeding it that could be damaged by air pressure being forced into its outlet. They're usually a rubber diaphragm servo action valve that is only meant to block pressure on the inlet side and probably won't take kindly to having it forced through them backwards. So, it would be best if you could arrange an adapter that would seal against the inlet seat and the top so the air pressure only gets applied to the outlet of the vacuum breaker.



Zone Control Servo action diaphragm valve.