It is highly recommended to take a tour through Jordan, instead of crossing by yourself. Tours leave from all over Israel.
Below is some information that may help you. More details on the MISTI Jordan site.
You will need European-style two-prong adaptors for Jordan. Remember that if you have an Israeli pre-paid sim card, you will not have cell service in Jordan.
Dinar
Stronger than the US dollar, Jordanian dinar are worth about 1.4 dollars each. Depending on where you go in Jordan, you may be able to get away with only spending US dollars and using your credit card, as is the case in Petra.
Petra
Located two hours from the border crossing in Eilat, Petra is the premiere tourist destination in Jordan. Carvings from various historical periods may be seen there.
It is highly recommended that you go with a guide. However, if you do not, be careful: Petra was re-discovered only recently, and the people that lived there, the Bedouins, struck an arrangement with the foreigners who wanted to show the area off to tourists. Only the Bedouins would only be allowed to sell inside the city, their shops wouldn not be interfered with, and they would take a portion of ticket sales. In return, they agreed to move out of the city.
As a result, Petra is extremely touristy in that as soon as you step inside, men aged five to fifty will attempt to get you to buy their wares, ranging from donkey rides to bangles to food. You can partake, but beware that tips are expected for most services, and the Bedouins will be aggressive in trying to get you to buy their wares. Be advised that it is free to get on a horse or a donkey or a camel, but it is not free to get off-- the person who guided you onto the animal expects a tip at the end of the ride.
At the Treasury, plenty of young men will offer to take you to the top in order to get a better view of the Treasury. This is incredibly dangerous and illegal.
Wadi Rum
The Wadi Rum, pictured above, is a gorgeous desert in the south of Jordan. Wadi means desert, and Rum means moon. And although the picture is, in fact, Jordan and not Mars, blockbuster films looking to shoot on "Mars" go to the Wadi Rum, such as the Martian, Red Planet, and Rogue One.
The desert used to be a sea bed millions of years ago, a fact that is reflected in the layers of the sandstone.
Jeep tours are an efficient way to see the desert in all of its glory and climb the mountains, which are surprisingly easy to scramble up. For these tours, hats are not recommended as they will blow off, either in the jeep or on top of the mountain.