There is a pipe connected to irrigation system right after backflow preventer and some sort of pressure control. It has a coupling that was most likely used to winterize the system. What kind of coupling is this? How do I connect an air compressor to it?
That's a quick coupling valve. They're often used on commercial or institutional jobs (large grounds) to provide a place for connecting a temporary hose where a permanent hose bibb would be undesirable. They're less common in a residential setting, so they're going to be hard to find in big-box retail or hardware stores. Try an irrigation supplier.
Yours looks a lot like the Rain Bird part pictured below. I suppose Hunter, Toro, Orbit, etc may also make a similar, but perhaps incompatible, part. Have a closer look at yours to see if you can find any identifying brand/logo/part number.
The mating part is called a valve key.
(Photos from sprinklersupplystore.com.)
I think your device is an atmospheric vacuum breaker that facilitates self-draining of the irrigation system via one or more automatic sprinkler drains.
Some sprinkler systems are designed in a way that there is no need to blow them out with compressed air. They use automatic drains (video here).
Water will not drain properly unless make-up air is allowed into the system. Your device, I believe, snaps shut when there is water pressure. When pressure drops to zero, the device opens to allow air to enter the pipe, which allows the drain or drains to empty the sprinkler heads and pipes.
You can test this theory by connecting compressed air using any fitting of your choice to the pipe where this device is / was connected. If you get very little water out of the sprinkler heads -- just a few spurts -- and mostly just air coming out of the sprinklers, then you have automatic drains and blowing out the system is unnecessary.
If you have automatic drains that are not working well and you want to blow out the system, you can add a T to the plastic pipe just below the device, insert a ball valve into the T and connect a male quick-connect air fitting. It seems an ideal place for it.