Born in Eschweiler, Altmeyer began his performance career while still a voice student at the Hochschule für Musik Köln where he studied under Clemens Glettenberg from 1953 to 1956. His first successes were primarily as an oratorio soloist. He was hired by the Westdeutscher Rundfunk (WDR) to perform in several recordings of cantatas and other religious music. He went on to win second prize at the WDR's singing competition in 1955. In 1956 Altmeyer joined the roster of singers at the Berlin State Opera, where he sang for the next four years. While there he notably portrayed the title role in the world premiere of Humphrey Searle's The Diary of a Madman. Then starting in the 1960-1961 season he was the principal lyric tenor at the Staatsoper Hannover, making his debut there as Tamino in Mozart's The Magic Flute. He became particularly successful in that house with several more Mozart roles like Ferrando in Così fan tutte and Don Ottavio in Don Giovanni. He also excelled in portraying roles in works by Albert Lortzing and Gioachino Rossini. Other roles of note included the title roles in Benjamin Britten's Albert Herring and Jacques Offenbach's Orpheus in the Underworld. He also found particular success in the title role of Hans Pfitzner's Palestrina, a role he also performed at both the Vienna State Opera and the Staatsoper Stuttgart