From Longman Dictionary of Contemporary Englishwalk-upˈwalk-up noun [countable] American English informal 1 DHHa tall building with apartments in it that does not have an elevator2 DHHan apartment, office etc in a building like this
Examples from the Corpus
walk-up• He lives on the fourth floor of a walk-up building.• In her walk-up, there would be a bed.• She lived in a red-brick tenement in Chelsea, an old walk-up building with gloomy stairwells and peeling paint on the walls.• The Kelly factor is big, and Rodman also had a big impact on our walk-up. 