Born in Vienna, Rappoldi played piano and violin as a child, and was already performing his own compositions at the age of seven.
After a time (1861–66) as concert master of the Deutschen Oper in Rotterdam he was Kapellmeister in Lübeck, Stettin, and Prague.
In 1871 he joined the faculty at the Königliche Hochschule für Musik in Berlin and played viola in the Joachim Quartet.
Rappoldi composed two string quartets, small symphonies and sonatas for violin and piano, as well as about 30 songs.
[4] The more famous Joachim Quartet performed the Beethoven cycle for the first time fifteen years later in 1903, and they never included the Grosse Fuge.