Three things move a short-notice booking fastest. Your cat's vaccination record, in whatever form your vet gave it to you: core feline vaccination (FPV, FCV, FHV) with proof of a booster or a titer result, or a vet memo if there is a medical reason not to vaccinate. Your cat's usual food, so a hard week does not also become a diet change. And any medication, with the dose and the timing written down rather than explained at the door.
Two more things save a phone call later. Parasite prevention is administered at check-in. And name a backup contact who can answer questions about your cat while you are in surgery, in the air, or otherwise unreachable.