A compression test, which, by the way, does not require a compressor, is perhaps the most common method used to diagnose engine condition, but the results won't necessarily indicate if the head gasket is leaking.
Compression tests are easy for most of us as they require minimal test equipment (just the compression test gauge, the car and a fully charged battery), but since you have an air compressor, a leakdown test might be more informative - do you have an autoparts chain nearby that rents tools?
The leakdown test set uses the air from the compressor and feeds a controlled amount of compressed air into the cylinders, one by one, through the spark plug hole, the gauges allow you to get a visual idea on how much air is leaking, and you can listen/observe to get an idea of where it's leaking - a hiss into the intake or exhaust would suggest the leak is past the valves, bubbles in the coolant could be a headgasket (or a cracked head/block) and so on.