How much does Luffy hate being a prince? And what’s his relationship to King Garp?

Its complicated - he hates the responsibility, loves the constant access to food LOL.... He really doesn't subscribe to traditions or customs or whatever, he plans to just do whatever he wants when he becomes king - he already does whatever he wants, or whatever he can get away with.

His relationship with Garp is extremely strained, just like in canon - Garp knows that Luffy will never live how he asks, and it drives him crazy. The absence of Dragon also doesn't help, Luffy isn't the first heir to the throne to completely disregard everything Garp stands for.