How to transfer frame of brood to queenless hive

I recently discovered a queenless hive. It has no eggs. I am going to transfer a frame of brood with a queen cell attached. It is a 2 hour drive from hive to hive.

Any problem transporting, temperature is in 70's?

How long does it take a queenless hive for workers to start laying drones?

Do you have a nuc box, so you can carry the frame with bees on it? Make sure the frame has eggs, so you know you'll have the youngest possible larva.
I've read laying workers start 3-5 weeks after the hive considers itself hopelessly queenless.