(Oxford English Dictionary)
walk-up, a. and n.
1. Of an apartment, etc.: that has to be reached by stairs rather than by a lift. Also applied to a building consisting of such apartments. U.S.
1919 MENCKEN Amr. Lang. iv. 110 The term flat 'is usually in the United States restricted to apartments in houses having no elevator or hall service'. In New York such apartments are commonly called walk-up apartments.
1927 E. GLYN It iv. 40 Mary had a tiny two~room apartment in a walk-up building in Brooklyn.
1942 N.Y. Times 13 Feb. (late City ed.) 13/4 Six five-story walk-up apartment houses.
1946 MEZZROW & WOLFE Really Blues (1957) xiii. 235 Some crummy walk-up tenement flat.
1979 H. KISSINGER White House Years xxi. 906 Huang Hua and I met around six o'clock in the CIA's walk-up apartment in the East Seventies.
1984 Business Rev. Weekly (Austral.) 4-10 Feb. 42/3 The Myer development is a three-storey walk-up affair that has appealed to a lower age group than usual for Gold Coast units.
2. That may be approached on the street, without having to go into a building.
1963 C. J. MCCALL in A. Dundes Mother Wit (1973) 421 The colorful window-signs of innumerable 'readers'..cry out from their store-front or walk-up locations.
1972 Sunday Sun (Brisbane) 8 Oct. 16/1 The same bandit had approached Miss Avery at her outside walkup window post on Tuesday and demanded $3000.
1. A walk-up apartment or apartment block. U.S.
1925 Scribner's Mag. Oct. 6/2 Vacation heaves into sight over the horizon..the swirling dust turned into clean sand; the only walk-up a dune; and the total night life two movie theatres.
1942 R. CHANDLER High Window xxv. 149 The kind of dentists who have shabby offices on second-floor walk-ups over stores.
1954 F. P. KEYES Royal Box xiii. 165 The friends he had all lived in the identical kind of six-flat walk-up.
1966 R. STOUT Death of Doxy (1967) i. 6 The person to ask lived on the second floor of a walkup on 52nd Street.
1976 National Observer (U.S.) 4 Sept. 1/2 The blue-jeaned couples climbing the stairs to their walk-ups together are most usually the children of affluence.
1980 J. KRANTZ Princess Daisy xxv. 438 Daisy herself lived in a low-rent SoHo walk-up and held down a full-time job.
16 December 1906, New York (NY) Times, "In the Real Estate Field," pg. 17:
Perhaps the most important fact brought out in this renting was that whatever oversupply of houseings there may be north of 135th Street is very largely in the cheaper "walk-up" flats, whereas tenants have been found readily and at good rates for accommodations of the better sort.