I will meet him . . . . . . . . Saturday afternoon.

Prepositions are words that are used to express temporal or spatial relations between people, objects, concepts, etc.
On is a preposition that is used as a function word to indicate contact or support with a surface.
It can also refer to attachment, dependence, possession or conveyance.
Here, on is used to indicate a space of time when something will take place.
In addition to being a preposition, on can also function as an adverb or adjective.