He was born in Selje Municipality as a son of district physician Johan Nicolay Cappelen (1818–1890) and Elisabeth Brunchorst Ravn (1825–1906).
His brother Christian was also a physician, and through him, Axel was an uncle of Minister of Justice Johan Cappelen.
[2][4] In 1889 Axel Cappelen was hired at Nordre Trondhjem County Hospital in Namsos.
After a period in the surgical department at Rikshospitalet from 1893 to 1896 he served as chief physician at Stavanger Hospital from 1897 until his death.
After two days of intensive care, the patient died of coronary occlusion and not because of the repairing of the heart.