Travelling from Amman to Cairo by bus is considered a rather uncomfortable and expensive way to transport and most people prefer to fly with the Royal Jordanian or the Egyptair Airlines. However, the ones that chose the overland option, they are prepared for a long and tiring 21-hour journey and it is suggested to have a break –stop at Aqaba for some breakfast, before moving on. In order to reach your destination you can use the services of JETT bus operator as well as other Abdali-based ones.
Note that flying to Cairo is more expensive than driving there by bus.
As for additional information upon transportation means, you can visit the: http://www.farecompare.com/flights/Cairo-CAI/Amman-AMM/market.html
Here you will find a list of reliable provider to book your ticket from Amman to Cairo: