Removing the airbox is a pain. So I took some shortcuts.
I already put foil tape on the damper doors of my '89 Jetta's airbox (the foam was shot). But, I thought there might be more old foam in the heater core, so I did a ghetto trick that worked well in my '86...
I drilled a few holes to get a compressed air blowgun aimed at the heater core.
A mini bar-clamp can hold the forward vent dashpot in the open position:
No pics, but maybe you get the idea. With the heat control lever in the full hot position, I reversed the blower fan leads and used the airgun through the holes to move loose debris to the evaporator while the fan was running in reverse..
Then, I restored the blower leads, flipped the dash lever to cold and used the blower fan to blow junk out the front vents.
I got some more junk out this way, but not a lot, since I already had cleaned it once. So, I taped over the holes and put the console back in.
I also saw this trick on another forum: put foam pipe insulation into the gaps in the grille.
Cut slits to get the foam past the vertical grille supports.
It's waterproof and the only tool you need is a knife. The grille is quite far from the radiator, so the fan can still pull air through the radiator if the engine ever needs it.
These pieces are from last year; I just got them out of storage and pushed em into place. I'm too lazy to make em straight & pretty, but they work.