there is a minibus type bus service that operates from Damascus gate in Jerusalem to Allenby/King Hussein Bridge crossing. I recall paying something like £3. Maybe a few more pounds now.
On the Jordanian side you will find many taxis to Amman. Some will probably also be there to take you to Petra, but I am not sure about the details. One thing is for sure you will not be stuck at the border. To Amman the taxi fare was about £20.
Crossing Israel to Jordan is fairly quick. Crossing Jordan to Israel can take some time depending on your ethnicity/religion and previous passport stamps.