Baumann was born in Seibersdorf Castle in Troppau, Austrian Silesia, and grew up in a middle-class environment, which enabled him eventually to study at the ETH Zurich.

After completing his architectural studies (1870–74), he was employed by Wienerberger Ziegelfabriks- und Bau-Gesellschaft beginning in 1876, and then by Viktor Rumpelmayer from 1879 to 1882.

Initially influenced by the international Renaissance revival due to his studies in Zürich with Gottfried Semper, around 1900 Baumann switched to a conservative Baroque-revival vocabulary as an Austrian "imperial style".

Baumann was also selected to design many buildings in the Lower Austrian city of Berndorf when it was significantly expanded under Arthur Krupp.

The injuries sustained could not heal properly, and Baumann spent his last three years in a wheelchair before dying in Vienna.