The only border crossing between Jordan and Syria is at Ramtha/Der�a. Ramtha, on the Jordanian side, is about 88 kilometers north of Amman, and Der�a is 100 kilometers south of Damascus.
Jordanian JETT buses, and their Syrian counterparts from the Karnak company, connect Amman with Damascus. The trip takes between five to six hours, including the time taken to complete the formalities at the border. Book at least two days in advance, and expect to pay around 5 JD. Servees taxis to Damascus run from the Abdali bus station.
There is a JETT bus service between Amman and Jeddah, Saudi Arabia, which runs three times a week. The trip lasts around 22 hours. You will need to get a visa for Saudi Arabia.
Likewise, JETT operates daily service between Amman and Baghdad, but getting a tourist visa for Iraq is quite difficult.