Will Golf Pad work without Internet connection?

Golf Pad saves course data the first time you play a course. So if you started a round while connected to the Internet, you'll be able to play that course even if there's no Internet connection anymore. Just select the course from the 'Recent' list.

What about aerial maps of the course?

Downloading and permanently saving aerial maps is against Google terms of use. The good news is that map tiles are cached automatically whenever you look at them within the app. So, if you start a round and browse course maps (just navigate between the holes in the map view) while connected to the Internet, then in most cases you will be able to see the maps during the same round, even if there is no Internet connection. 

Also see this article for the estimated data use when you play the golf round.