1834, "an act of running upward," from verbal phrase (late 14c.), from run (v.) + up (adv.). Extended sense "period of time or sequence of events proceeding some important event" is from 1966.
1833, "customary track of an animal," especially a deer, American English, from run (v.) + way (n.). Meaning "artificial sloping track" is attested from 1883; airfield sense is by 1923.