Bathroom exhaust vent is on the ceiling and quite difficult to reach as it exactly above the wash basin. I wanted to clean it up as it was dirty. I used vacuum cleaner and then wiped with wet sponge, but the result was not effective. I could not clean thoroughly and could still see the dusts on the inner sides. Any hacks for this without removing the mesh from ceiling?
Take a ladder, go up and use a toothbrush. Rub for some time and then rub with wet sponge. Do this until you clean it from the outside as much as you can and then use a vacuum cleaner.
I don't see very well what kind of cover is on the fan, but sometimes if you can't pull out the dust then you can try to blow it in. If your vacuum cleaner can't blow (I mean if it can't reverse the air flow) then use something else which produces air under pressure, for example a can with compressed air for cleaning computer keyboards.
As some of the dust may be statically clinging to the vent, perhaps use an 'air duster' to loosen the dust, such as the one below:
As you do this, have the vacuum handy to suck up the loosened dust.