In a nutshell, Duke Paolo is Prince Fabrizio's spoiled, good-for-nothing son. He's lazy, entitled, and bitter about the fact that his family is going to lose its status when Sicily becomes part of modern Italy. When he finds out about his cousin Tancredi fighting alongside the invading soldiers, he complains, "[He's] gone to join those swine who're making trouble all over Sicily; things like that just aren't done" (1.128). Paolo doesn't know how to argue against something other than saying, "thing like that just aren't done." Nice airtight logic there, Paolo.
On top of all that, Paolo is jealous over his cousin Tancredi's success. He knows that Tancredi is better suited to the new political world, but he tries to hide his jealousy by criticizing Tancredi's political loyalties. Fabrizio sees through this guise immediately and thinks, "Personal jealousy, a bigot's resentment at his agnostic cousin, a dullard's at the other's zest, had taken political guise" (1.129).
Paolo's not fooling anyone. At the end of the day, he demands all the respect and luxury in the world simply because he was born into the right family. But he's about to get a firm reality check when democracy comes to Italy.