The vet won't give your cat anything. And it is very highly discouraged in the first place. Any drugs will slow the heart, which also happens when you are exposed to less oxygen (ie-when you are flying). Thus, it compounds the slowing of the heart which can then cause it to stop. I highly recommend NOT drugging your cat in any way shape or form.

If you are going further west than about Chicago/Dallas, be aware that they may not fly your cat all the way there in one day. I was on the phone with every airline that flew into Albuquerque and not one could get my cats to Switzerland in one day (thus, coming the other way is the same). They have regulations about the number of hours an animal can be flying, minimal connection times for its transfer at an airport, etc... This only applies if it is not in the cabin with you but instead in the cargo hold. I recommend that you book the cat's ticket first, or rather, confirm a set of flights your cat can take (by calling the special pet telephone number at your favorite airline), and then book the same ones for yourself.

And just as an FYI, the price for traveling with a pet has definitely gone up from what is listed here (each airline charges differently which is why I don't list specifics).