If cleaning a saw for a rebuild, or to put up for a few months, I don't see any way around using lots of soap (Simple Green works ... cheap at Home Depot) and lots of hot water from a high speed high volume hose nozzle. To me, the detailing methods--like air needles etc, would be like trying to pick a thousand little buggers out of the saw here and there. And then the little sawdust buggers get everywhere in the shop. It's an outside job. The grunge, the hardened sap/oil/polymer impregnated sawdust, in the carb box, covering the flywheel and ignition, is typically too thick and too everywhere ... so, for me, it's outside with soap and hot water, and later it's soap and an old paint brush and a laundry tub for detailing parts.)
Once you decide to use soap and water to clean the carb box (and I don't see a way around it for the hardened-up grunge that gets caked in there) you'll need to remove the air filter and cover the inlet opening to prevent high pressure water from getting into the passage way and then into lower crankcase ... take 5 minutes (oh darn, there goes my whole day again) and cut a small piece of 1/2" wood to fit as a cover where the filter once was.
(IMHO, air filters which screw in, are a net advantage due to ease of cleaning. I don't see where the snap in filters are a net advantage. For a snap-in filter, I haven't a clue as to how to secure the opening with the filter removed, and you have to remove the filter to really expose the carb and carb box for cleaning. I don't think jamming a piece of oil rag or other plug jammed to cover the venturi would survive the high velocity water pressure which is needed to clean the gunk out of the box. Maybe jam a wad of oil cloth into the carb inlet, and then replace the snap in filter, and then do you best to clean under the filter. That's the advantage of the screw-in air filters--you can remove 'em and screw in a cover plate.)
Same thing goes for the muffler. Once you decide that an outside job, with soap and hot water, is simpler and faster than detailing with air inside, and then cleaning up the saw grunge which gets sprayed all around shop, then you're going to need to keep water from getting inside the muffler opening, as it will settle in the bottom of the muffler and then rust the muffler from the inside out. So, that means pull the muffler, and that then also means, make another wooden cover for the exhaust port (Oh darn, there's another 5 minutes with a 1/2" piece of wood and a hack saw--there goes my whole day ... again)
Blast away with hot water ... if using one of these dedicated high speed hose jets (they are amazing, and the standard hose nozzle is no competition -- that little thing will move bricks), you'll need to stand back about 15-20 feet or so, or the backspray will have you dripping in Simple Green, saw grunge and hot water in a blink. Champher the nozzle inlet 1/16" (not shown) to increase velocity and volume by maybe 50% (shifts the vena cava of the nozzle back toward the source.) 10 feet will be too close.
Clean the airfilter with an old toothbrush - per owner's manual, probably daily (oh darn, that much effrot just about kills my whole day--all over again! Do I really have to wipe my own a$$ each time ah takes me a dump?)
Sounds like overkill, but it's actually a good question. This is as simple a process as I could hone down, it cleans the saw good, and it keeps sawdust grunge out of the basement/workshop.