Air hose if you have one, so far that is the best way to clean our articial tree. We also store our tree covered in a big garbage bag or plastic hanging upside down from the garage ceiling. This has really kept it pretty clean, plus we get great comments on how our tree "hibernates" or looks like big christmas tree bats hanging in our garage.
Before I put it away after Christmas I use the fabric/upholstery brush attachment on my vacuum to clean it. Start at top and work your way down. Sometimes if I notice it is really dusty I will use a damp cloth to wipe down all the branches. When I put it away wrap it in bed sheet, that helps keeping it clean.
Good luck! I love Christmas time!
This website has some great instructions: http://www.wikihow.com/Clean-an-Artificial-Christmas-Tree
Yes the air compressor will work, just be careful about the pressure of the air coming out otherwise you may start to lose the greenery off the tree.