Just replace the foam filter every few years. The oil breaks down the rubber foam...
Almost any foam rubber will work. You can run it with no filter but then risk getting big particles in the carb or the cylinder which is bad.
You could rig up a paper filter or rig up a bigger filter box attached to the air intake, but much easier to just replace the foam every few years.
You need to be starting the generator more often than yearly just to let it run.
Also, you want to run 100% pure gas in it, not any ethanol which breaks down, contains H2O and will harm the carb if it sits for any amoutn of time, though running on ethanol mix will be fine as long as you store it with 100% gas having been run enough to get the carb to contain 100% gas.
**edit. Neighbor gave me his chainsaw and I cut 12 trees... Finally got tired of carb being out of adjustment, so took cover off to work on it and no filter at all. Yes, motor will run without filter (might cause the carb to be out of adj since it no longer has to pull air as hard through the filter) but you do not want that junk in the carb, or any scratchy materials to make it into the cylinder and scratch the piston...**
