My first choice would be to drive if your situation allows for that. Twelve hours seems like a long time but it's doable. In the worst case, as you mention, you can spend the night. The puppy will be so discombobulated he/she likely won't be all that much trouble. It will likely be a bonding experience.
You could also fly if by that you mean fly out and then back with your bundle. You should be able to take them in the cabin with you. That is also doable but it means having a stranger (you) take the puppy out at security. That might prove problematic or it might be fine. Flying one way and driving back would also work. You could fly in, meet your darling, and then the next day pick him/her up and drive home.
I haven't used specialized ground transport but if the reviews are good and you can trust them I can see that working.
How old is the puppy?