Because I live in a coastal area, and have worked in marine environments, I became aware that if screws holding the fan in place have been there for a long while, galvanic corrosion between the aluminum fins and the (usually) steel or brass screws can make them difficult to extract. I usually remove the screws first chance I get on receipt of a system, and replace them with new fasteners (unless they're still really new).
If the fan is pulled at every maintenance interval (I go about six months), the fasteners generally don't have time to corrode unless you live near a body of salt or brackish water.
If you can get hold of some, medical swabs (which have less cotton on the tips than regular Q-Tips, and have a long, slender wooden shaft) are excellent for all sorts of detailed cleaning work. (I haven't seen any on sale in the pharmacies around here... only at the few remaining electronics parts stores in the state). They have the reach to get in those deep places without disturbing stuff too much.
I figure nobody here needs it, but I'd feel remiss if I didn't remind everyone to only do work cleaning inside a computer's case with the system shut down, the power off, the cord unplugged, and the power button pressed down for several seconds after unplugging (check for a standby LED on the board and make sure it's out). If it's a laptop, please also pull the battery, even if it's known to be "dead". Safety First.