Brost became friends with singer Mick Jagger after they met in 1965 when Jagger and the Rolling Stones performed in Malmö and came to a nightclub where Brost was working. When told about Brost's death, Jagger stated that "He was a wonderful guy" and that "We will miss his humour" and his generosity.
As a child, he did not want to have the same job as his mother, who he sometimes accompanied to her work as an Actress. However, he later chose to start acting anyway and applied for the Royal Dramatic Theatre drama school, but failed to make it through the first round of auditions. He took private acting lessons with Director Fred Hjelm, and then applied for the Teaterhögskolan in Malmö, where he was accepted. He graduated in 1970. After graduation he was employed by the Stockholm City Theatre, where he stayed for two years. After that he acted at Åbo Svenska Theater in Finland, and then at Riksteatern, Unga Teatern and Malmö City Theatre. He also took part in the Tältprojektet in 1977.
In 1981 he had the leading role in the SVT Christmas calendar titled Stjärnhuset, which won him national recognition. A year later the ratings success Gäster med gester began airing on SVT and Brost became a panel member and showed off several pantomimes. He also participated in several comedy plays such as Bäddat för sex and Är du inte riktigt fisk at Chinateatern, Me and My Girl at Intiman in Stockholm and Kuta och kör at the Palladium in Malmö.
Brost perhaps became best known for his role as Joker the bartender in the television series Rederiet, where he appeared in the entire 318 episodes of the series between 1992 and 2002. He also had roles in several film such as Black Jack, Änglagård, Jönssonligan dyker upp igen and The Visitors.
In 2013 Brost won a Guldbagge award for his leading role in the 2011 film Avalon. He also played Pekka in the series Jordskott and appeared in the show Lilyhammer.
In 2016 Brost took part in the SVT show Stjärnorna på slottet, where he spoke about his life and career. He also appeared as a Celebrity Dancer in Let's Dance 2017 on TV4. In December 2017 he had his last role in the SVT series Fröken Frimans krig, playing Sievert Lindberg, a wealthy man interested in a friend of Dagmar Friman.
Brost had four children. He died from complications of throat cancer on 4 January 2018; he was to appear on a TV4 program, Tillsammans mot cancer (Together Against Cancer) on 8 January.
