What requirements should I look for in an air compressor to clear irrigation lines?
CFM - cubic feet per minute - is what rushes through the pipes and carries out droplets of water. PSI - pounds per square inch - is what turns ordinary sprinkler heads into high-flying air powered rockets. No kidding.

How do the big pros do the job??

In my part of the USA the contractors use towable air compressors to sprinkler blow-out. I own two of these and rent them out each fall. Mine are in the 175 CFM class -- they're the type you'd see a road crew using to power a jackhammer for instance. Although these compressors can produce over 100 PSI I prefer to turn it down to about 60 PSI. With a relatively high-flow source like this I can easily blow out a quarter acre residential sprinkler system in 20-30 minutes, including putting the air hoses away.

OK, I'm not gonna buy nor rent a towable. It's fine if the job takes me longer.

There's a tradeoff between compressor size and job speed. Before I had these compressors I used my electric compressor, an upright unit with a 60 gallon tank and 240 volt motor. I don't recall its CFM spec, but after the tank de-pressurized the compressor pump could move enough air continuously to get the job done at least halfway. I'd go through every sprinkler zone twice to make sure I'd evacuated the water thoroughly. The job took an hour or more.

My brother used to use his portable compressor to do the same. It was big as portables go, maybe 15 gallon tank at the most. Sprinkler blow-out usually took him a couple hours because he'd have to blow off each zone several times and it took several minutes for the air compressor to recover after each shot of air. The biggest difference between his and mine is not actually the size of the pump, though -- it's the size of the air tank. Because my tank stored more air it could push more water further with each cycle. The towables get the job done so fast with no tank at all: their pumps move so much air instantly that a tank isn't needed.

Can I set the air pressure at max?

People are often set the regulator at a higher pressure. This can help an undersized compressor get the job done by packing more air into the sprinkler main line, acting as if the compressor itself had a larger tank. This does increase the risk of damage to the system however. Valves open with more force, zone piping takes more shock when the valve pops open... and yes, sometimes sprinkler heads really do get launched into the air. Most of the time there's no damage, though, so many people accept that risk.

Finally, a recommendation!

If the property you're going to winterize is not very big, say under a quarter acre, then there's less volume of pipe to be evacuated. If you don't mind spending half a Saturday doing the job, that's good too. I'd suggest you'll want something with at least an 8-10 gallon tank since that stored air is what gets the job done. 15-30 gallons tank volume would be much nicer. The CFM of the pump just controls how long you'll have to wait before there's enough air to blow it off again.

One thing you can do to expand tank capacity is to get a portable air tank (or a broken compressor). This may be more economical than buying a compressor with a larger tank, especially since it sounds like you may not use it heavily.