You have to pay the Israeli exit tax when you leave for Jordan. You can pay at the crossing or pay in advance:
- Any postal bank branch in Israel, with no extra charge*.
- the "Milgam Co." website, by credit card.
All intercity buses in Israel have large luggage bins.
You should book bus tickets to/from Eilat:
http://www.egged.co.il/Article-809,1...-to-Eilat.aspx
You can see the Jerusalem-Eilat schedule on the same website.
If you decide to fly, look at the "Arkia" and "Israir" websites for schedules and fares. If you fly back to Tel-Aviv it is more convenient to fly to Sde Dov airport (and not to Ben-Gurion).