I traveled ten hours from Oregon down to Cali in a car. I actually just left my Betta in a Petco PetKeeper container, and she did fine. We did have to make sure that the water wouldn't come out whenever we ran over a bump though, and I lowered the water level to about 3/4 the usual amount.
I did not have packing material on hand, but I believe the best way to pack a fish is the way they come packed in fishstores or from Aquabid. Basically the thick plastic bag filled halfway with water and packed tight with air to offer some cushioning. You can also put the plastic bag in a box with some insulation to protect your Betta from temperature changes. Also, it's a good idea to fast the Betta a day before traveling if you're transporting it in a small amount of water, since no food = no waste production = better water quality.
The Betta I ordered from Aquabid took about 2-3 days traveling through USPS to arrive to me. They all came in very lively and healthy. 22 hours in a car with you while you care for the fish should be nothing compared to how USPS might handle the package. So don't worry!