Having premiered many solo and chamber music pieces by modern composers and performed as a soloist with renowned orchestras such as the Danish Radio Symphony Orchestra and the New Taiwan Symphony Orchestra, he is known for his versatility in both traditional cello repertoire and contemporary compositions.

Due to his experimental approach to classical music, he invented the "Bach Ultramarathon," in which Møldrup plays the 36 movements of Bach's 6 Cello Suites and runs one kilometer between each movement. This event premiered in 2022 at the Bergen Festival in Norway, the Frederiksberg Festival in Copenhagen, and the Artlink Festival in Belgrade.
Møldrup has received numerous prizes and awards for his achievements in music, including the Queen Ingrid Honorary Prize and the Augustinus Foundation Jubilee Scholarship.

Møldrup is currently an associate professor at the Royal Danish Academy of Music and directs the two annual music events “Svanekegaarden Cello Masterclass” and the “Fynposium Festival”.

He is a popular masterclass teacher who has taught at institutions such as the Manhattan School of Music in New York, the Royal Academy of Music in London and the Shanghai Conservatory of Music in Shanghai.

He was principal cellist of the Copenhagen Philharmonic Orchestra and a member of the acclaimed Paizo Quartet and the Danish Piano Trio, and has made numerous recordings for various labels, including Dacapo Records, Bridge, and BIS Records. He plays a David Tecchler cello from 1697, kindly loaned to him by Augustinus Fonden.