In "stick something on", is "on" an adverb or a preoposition?

to stick on is a phrasal verb, so it's "part of the verb". Context will define if it's serving as an adverb or preposition.

=> Stick it on. (adverb in this case)

=> Stick it on his head. (preposition in this case)

To stick on (to adhere)

To stick up (someone) (to rob)

To stick out (to be conspicuous)