To travel from Dubrovnik to Opatija with bus, you have the following options.

1. Take a bus from Dubrovnik to Split, from Split you then take bus directly to Opatija or to Rijeka from where there are local buses to Opatija
2. Take bus from Dubrovnik to Karlovac, In Karlovac change to bus bus directly to Opatija or to Rijeka.