Enjoy your stay in Israel. You can visit Jordan through the bridge on the border, then you can travel all the way South through Amman to reach Petra that will take:
2-3 hrs fro Tel Aviv to Amman
5-6 hrs from Amman to Petra.
Don't forget to buy a map from any book store in Jordan to recognize names and places.
Have fun
You will love it.
P.S: * Crossing to/from Israel and the PNA Territories
There is one direct crossing point between Jordan and the Israeli-occupied West Bank: the King Hussein Bridge (also known as the Allenby Bridge). Since Jordan and Israel signed a peace treaty in October, 1994, two new border crossings have been opened between the two countries. The Sheikh Hussein Bridge is at the north of the Jordan Valley, and the Wadi ‘Araba crossing is in the south near Aqaba and Eilat. Each of these entry points are open from 08:00 until 23:00 from Sunday through Thursday, and till 14:00 on Fridays and Saturdays. They are closed on most Jordanian and Israeli holidays.
Leaving from Amman to al-Quds (Jerusalem) through the King Hussein Bridge, you can either take a JETT bus (tel. 5664-146), which leaves at 06:30 daily, or a servees or minibus from the Abdali bus station. Once at the terminal, you will be checked out and board another JETT bus, which is the only form of public transport allowed to shuttle across the bridge. Once across the bridge there is an Israeli passport check before you arrive at the immigration terminal on the West Bank.
There your luggage and person may be searched, and you will receive an entry stamp. Although this stamp does not present a problem when entering Jordan or Egypt, if you wish to visit Syria, Lebanon or any Arab country which has not signed a peace treaty with Israel, you should explicitly ask to get your Israeli entry and exit stamps on separate pieces of paper, not in your passport. Likewise, you should ask the Jordanian authorities to stamp your exit and entry from any crossing point connecting Jordan with the occupied West Bank or Israel on a separate sheet of paper as well.
After being checked in by the Israeli guards, you can catch a share taxi to Jericho, Jerusalem or other places on the West Bank. Buses are also available for a lower price.
Coming from East Jerusalem to Jordan, you can catch a taxi just opposite the Damascus Gate for the King Hussein Bridge. The trip takes 40 minutes to the bridge checkpoint on the Israeli-occupied side. After paying an exit tax and being checked out, you will board the JETT bus to shuttle you across the bridge to the terminal on the Jordanian side. After you are checked into Jordan, board a servees or the cheaper public minibus that will both drive you straight to the Abdali bus station in Amman.
Jordanians and foreign visitors are allowed to drive their cars into Israel and the West Bank provided that they hold a driver’s license and a car license translated into English, automobile insurance (which can be issued at the borders) and an entry visa to Israel if necessary. Bus transport is also available between Amman and Tel Aviv, between Irbid and Haifa, and between Aqaba and Eilat.
Getting Around in Jordan
Taxis are available from Queen Alia International Airport to take you into Amman. The trip takes around 30 minutes, depending on traffic, and will cost you around 10 JD. The airport also has a bus service which arrives at Abdali bus station every hour and costs 0.500 JD. Passengers using Queen Alia airport for international flights are requested to check in two hours before departure.
http://www.kinghussein.gov.jo/facts2.html