Hopefully you’ll see a rabbit-savvy vet and go to a clinic has dealt with bunnies before, so they know just how skittish a bunny-patient can be. It’s good if you can hold the bunny if he’s scared – a lot of bunnies if not most are scared at the vets. Bringing your sister is a good idea.
As for the check up: I think they will listen to his heart and lungs, inspect his incisors (the front teeth), perhaps feel him all over, look so his eyes are clear (no boogers or discharge), perhaps they’ll weigh him. And they’ll probably ask you if you’ve any concerns or if you’ve noticed anything that seems off etc. They might ask you about his diet and tell you he should eat lots of hay (which he should, but all rabbit-owners don’t know that).
Most buns are pretty disapproving of their hoomin after a vet visit and he might not want to “talk to you” for a couple of hours or even a day. As long as he’s eating and pooping, that will go away and he’ll start to trust you again. If you have some extra yummy treats, you can bribe him with those.