He was the son of Laurits Christian Frederik Michael Brandstrup (1812–1900) and Johanne Kirstine Fenger (1820–98).

Brandstrup attended Sorø Academy before training for five years as a carpenter with Severin and Andreas Jensen in Copenhagen, after which he spent a year studying in the sculptor Vilhelm Bissen's studio in 1884 where he learnt the art of sculpting marble in the Thorvaldsen style.

He then spent a short period at Copenhagen Technical College from where he entered the Royal Danish Academy of Fine Arts in 1885, graduating in 1888.

In 1899, he was awarded the Thorvaldsen Medal for the Equestrian statue of Christian IX which stands on the central square in Esbjerg.

[1][2] Other important works include statues of Ottilia Jacobsen (Glyptoteket, 1905), the jurist Georg Morville in Viborg and the astronomer Ole Rømer at the Technical University of Denmark in Copenhagen.