Time wise I would definitely fly between Cairo and Amman, doing that part overland is just too time consuming. Jerusalem can easily be done overland from Amman. And obviously Petra as well. If you fly into Amman you could visit Petra first and then head back north, take the King Hussein Bridge and visit Jerusalem. Come back to Amman via KHB on the same visa you got upon arrival at the airport or if it's cheaper to fly out of Tel Aviv to Cairo then that would be an option as well.
The entrance of Petra is a 2 kilometer walk down the hill, some of our guests enjoy the hike down but a taxi is very easy to arrange as well and for guests who like to start very early I drive them down to the site myself. There are many restaurants near the entrance of Petra so most of our guests have dinner before they come back up to the B&B. I'm happy to recommend a few good restaurants.