Several times weekly there is bus from Slavonski Brod to Bern, the bus travels via Zagreb and Zurich. On weekdays where there is no departure from Slavonski Brod, you can travel to Zagreb on your own and then catch the daily bus to Bern from there on. The bus trip from Slavonski Brod to Bern takes about 17 hours, the trip is conducted with modern long distance buses.
The bus departure from main bus station in Slavonski Brod, you can see address here, ticket prices and details about the travel route you can see below.