The son of organ builder, piano tuner and music teacher Franz Jacob Brambach, he was born in Oberdollendorf, a village across the river Rhine from Bonn. Brambach's mother, born Lückerath, was the daughter of a free-roving puppet master builder. His brother was German musicologist Wilhelm Brambach.
The young Caspar Joseph spent his first years in his native village, where he received his first music lessons from his father, which continued after he passed in Bonn elementary and high school and at the Conservatory of Music, Cologne. After that, Brambach followed his musical career as first violinist of the Bonn Opera House (1847–1850) and studied at the Cologne Conservatory (1851–1854), which promoted young composers and where he received awards for a string quartet and various songs for the Mozart scholarship at Frankfurt Liederkranz. Later he studied there as a private pupil of Ferdinand Hiller and Carl Reinecke before he himself became a teacher in 1858.
In 1861 he was appointed municipal music director in Bonn, where he led performances of oratorios by Bach, Haydn, Handel and other composers. He retired from this position in 1869 to devote himself entirely to his compositions and the musical life in Bonn. From 1862 to 1877 he conducted the men's choir "Concordia" and from 1861 to 1869 the Municipal Choral Society, now known as the Bonn Philharmonic Choir.
On 20 June 1902 Brambach died in his home in Bonn. His funeral was accompanied by singers from all over Germany. With great sympathy of the German minstrelsy, two years after his death a memorial designed by the architect Karl Senff was built on his grave at the cemetery in Bonn-Poppelsdorf. The relief on his donated, well elaborate tomb is signed by the Bad Honnef sculptor Charles Menser (1872–1929) and bears the inscription "Dedicated to German singers". It is now a grave of honour, located under the Cross Mountain Church.
In the obituaries of his time, the importance of Caspar Joseph Brambach as a musician and his work in the city of Bonn was highlighted. It was also added that he was one of the kindest and most unselfish people, whom no one will forget.