Set it on the counter upright.
If you put it in the mug hair down it will get a bend in the hair.
Some folks hang em, some stand em up...it matters not, but don't put em hair down in a mug.
Rinse well in warm water, squeeze, shake out most of the water and put it in an airy place.
Shake all the water out of it and either stand it on its base, or hang it in a brush stand bristles down. Either way that will allow it to air dry, and you won't have problems with it.