Mønsted was born at Grenå, Denmark. He was the son of Otto Christian Mønsted and Thora Johanne Petrea Jorgensen. His father was a prosperous ship-builder. At an early age, he began to receive painting lessons at the art school in Aarhus. From 1875 to 1879, studied at the Royal Academy of Fine Arts with Niels Simonsen and Julius Exner. In 1878, Mønsted studied under the artist Peder Severin Krøyer. In 1882, he spent some time in Rome and Capri then, the following year, visited Paris, where he worked in the studios of William Adolphe Bouguereau.
In 1889, he went to Algeria. Three years later, he travelled to Greece, where he was a guest of King George I who had been born a Danish prince. While there, he also did portraits of the Greek royal family. After that, he visited Egypt and Spain. 
During his later years, he spent a great deal of time in Switzerland and travelling throughout the Mediterranean His travels produced numerous sketches that became paintings he presented at several international exhibitions. Most of his landscapes were devoted to Scandinavia. He was especially popular in Germany, where he held several shows at the Glaspalast in Munich.