It doesn't have to be there, BUT you'd have to say "Pamatuji SI jeho slova." You get to choose either PAMATOVAT NA or PAMATOVAT SI (both followed by the accusative case). There is a slight difference but nothing you'd have worry about before you reach like C1. :D I'd say that the second variant is more common (but than again, this whole sentence sounds kind of bookish, so PAMATOVAT NA fits a bit better the style).
"pamatovat něco" without a preposition often conveys the meaning being old enough to be able to remember something in addition to just remembering.
Why is it "jeho" and not "něho"? I thought "jeho" goes without preposition, so i'd have said that after "na" comes "něho"
Jeho in this sentence is a possessive pronoun, not a personal pronoun. It is "his" here, rather than "him." Those little "n" changes happen with the personal pronouns.