I concur with PJRZ's answer regarding initiative and surprise. (In sum: the surprise rules can handle the question of whether the character and the rat are aware of one another and how, if at all, their actions in combat are affected as a result.)
However, I do think there would be a Dexterity (Acrobatics) check required for the character to avoid falling prone.
Since the jump in question (over a crate and onto a creature behind it) is intended to clear horizontal distance, not just vertical distance, it would have to be a Long Jump. The paragraph on Long Jump on PHB p. 182 states: "When you land in difficult terrain, you must succeed on a DC 10 Dexterity (Acrobatics) check to land on your feet. Otherwise, you land prone." And from PHB p. 190: "The space of another creature, whether hostile or not, also counts as difficult terrain."
Landing in the giant rat's space means landing in difficult terrain, and so would prompt a roll to avoid falling prone.