Kunz studied in his native Vienna, primarily with Theodore Lierhammer at the Vienna Academy. His debut took place at Tropau in 1933 as Osmin (a part for deep bass) in Mozart's Die Entführung aus dem Serail. Following that, he sang with a number of smaller German theaters before being engaged by the Breslau Opera for three years. Kunz made his first acquaintance with England when he was offered an opportunity to understudy at the Glyndebourne Festival in 1936. He was soon thereafter assigned several smaller roles. In 1941, Kunz became a part of the company at the Vienna Staatsoper where he remained throughout his career; he was given the title of Kammersänger in 1948. During the war years, he sang throughout Austria and Germany, primarily in Mozart and Wagner. He made his debut at the Salzburg Festival in 1942 as Guglielmo in Così fan tutte and in 1943 became the youngest artist ever to have appeared in a major role at the Bayreuth Festival when he sang Beckmesser in Die Meistersinger.
Once the hostilities ended, Kunz's career assumed a still more international flavor. Opera performances took him to Florence, Rome, Naples, Paris, Brussels, Budapest, and Buenos Aires. His role at the Salzburg Festival grew and he was a part of the Vienna Staatsoper troupe touring England and France in 1947. The following year brought his debut at the Edinburgh Festival.
A Metropolitan Opera debut waited until 1952, but Kunz's appearance as Leporello on November 26 brought a warm response from the audience and positive reviews from the critics. Both local and national writers commented upon his handsome voice and subtle comic skills. Many could recall only a few comparable artists in a role frequently immersed in slapstick routine. The Metropolitan Opera enjoyed his presence for just two years. In addition to Leporello, Kunz appeared as Mozart's Figaro, Beckmesser, and Faninal in Rosenkavalier. Chicago heard his treasurable Harlequin in Ariadne auf Naxos and Leporello, both in 1964 and, two seasons later, his wily, yet innocent Papageno in Die Zauberflöte. While musical tastes had moved from the elegant Mozart style of post-war Vienna to an earthier, more robust Italianate approach by the 1960s, Kunz's inimitable stage persona lost nothing of its potency. Nor did his voice; he continued to sing well even in his sixties and continued to undertake small roles (unforgettable cameos, all) to the end of a long career. In addition to opera house appearances, Kunz graced the stage of the Vienna Volksoper from time to time, giving lessons to both audiences and fellow artists in operetta style and singing.
Among the recordings of lasting value Kunz made during his prime years are, besides Meistersinger (two live from Bayreuth), Le Nozze di Figaro, Die Zauberflöte and each and every one of his operetta discs on Angel Records/EMI.