The song is written in the key of E Major set in a 4/4 time signature at a moderate tempo of 108 beats per minute.
Musically, "Run Up" is a dance-hall song.
PartyNextDoor and Minaj sing using the Jamaican Patois accent.
Reviewing the song, Ryan Reed from Rolling Stone described it as "a suave blend of high-pitched vocal samples, warped guitar chords and a signature Caribbean music groove."
Kat Being of Billboard call it as "a great follow-up to "Cold Water" (2016).