No, you will need much more volume than a home air compressor can provide.
The compressors we use are large, tow-behind the truck, gas engine type compressors.

I realize a big compressor would be best. But I think I can get by using a smaller one once a year.

My irrigation system puts out about 8 gallons per minute which equals about 1 cfm. I think the spray nozzles restrict much of that flow. A home compressor rated at 3 cfm and a 15 gallon tank should work just fine. I might have to blow out each zone a few times but I think it will work. Any thoughts on this?

Once the heads start blowing air, your compressor will lose the ability to keep up. Here's what I would do. If your system has a master valve, open it manually and start draining a station. Now once you get the water out of the main you can use it as an air tank. Keep the compressor charging the main with no station valves running until the pressure builds up. Then when you open a station valve, the air built up in the main will blow the station line out at maximum volume. You can do that as many times as it takes to get all the water out.
I have several systems with many stations well over 100 gpm and I have to do this every year.