In your first example, the phrase "on Friday" is a prepositional phrase that adverbially modifies the verb "meet." Likewise, the phrase "last Friday" in your second example adverbially modifies the verb "met."
Let me provide some examples of my own:
On Friday, I met him.
Last Friday, I met him.
Friday, I met him.
All three of these example sentences use an introductory phrase or word to adverbially modify the verb "met" in the main clause. Restructuring these sentences so that their respective modifying phrases aren't introductory or separated by a comma doesn't change the function of the phrases from being adverbial, for example:
I met him on Friday.
I met him last Friday.
I met him Friday.
Remember, an indirect object modifies the verb; ergo, indirect objects are adverbial. In all of these sentences, "Friday" is the indirect object of the verb "met" -- just like in your question.