Staying in Isarel after your trip might be a little complicated because Passover starts on March 24th in the evening. Hotels are probably booked up and where there's a vacancy, prices go up since many people will arrive a few days early.
So you'd have the 22nd which is a Friday, Saturday, Sunday, and Monday which is a half day of craziness as everyone needs to get somewhere.

Jerusalem would be frustrating since you really only have Sunday, and that day will be hectic anyway.

Maybe Tel Aviv, but do you have a place to stay?
Eilat is always expensive and since schools in Israel are on vacation starting the week before Passover--yeah, people mob the town and push up prices even higher starting then.

As far as Jordan, I haven't been there but plenty of Israelis travel outside of Israel for Passover too, so I imagine that the border crossings will be busier than usual. You can also expect that there will be a heightened security alert in effect (most majot holidays) so getting in or out of Israel might take longer for that reason too.

I'm sure your Birthright trip will be great, but on relatively short notice for a holiday season, I'm not sure how easy the after trip part will be.

WOW! Glad you brought Passover up, it hadn't even crossed my mind! Does anybody have experience crossing the boarder/flying out of Israel during Passover in addition to my other questions?

Can you check this form
http://www.fodors.com/community/africa-the-middle-east/egypt-highlights-2.cfm

We were in Israel for Passover 3 years ago. We specifically chose to stay in Tel Aviv during the first few nights/days of Passover because we figured it would be a bit less hectic. We arrived from Eilat in the afternoon of the first night of Passover. Things were still bustling and busy, but by about 3pm, everything had closed and no one was out on thestreets as everyone had headed home. Tough to find a palce to eat that night, but we were able to eat at the Hilton Hotel - though a very limited menu, but the food was good. The next day, most things were closed but there were a few more restaurants open and we had an excellent lunch at Goocha. It was a great day to just stroll the Promenade and relax on the rooftop terrace of our hotel. By the next day, everything was open and back to normal.

The weekend before Passover we stayed in eilat and it was very busy as families were arriving for the holiday. It is a bit upscale, though I found it a bit "Vegas", but there are a few hostels so you may be able to find something budget. However, if everything is already booked up you may consider crossing to Aqaba and staying there - shouldn't be as busy.

If you want to go to Petra and Wadi Rum and then fly out of Amman I would suggest you head south and do the border crossing in Eilat. From everything I've heard it is a much easier crossing than the one near Jerusalem. Then you can do Petra and wadi rum and head up to Amman from there. We did our Petra tour with Alahan Olympus (http://ahalanolympus.com/). We were very happy with them. We did a private tour but I know they also do group tours and in addition to Petra I know they go to Wadi Rum and Amman. Not sure how budget they are but atleast a place to start.

Here is a link to my blog which includes my full trip report of our trip along with pictures:
http://www.fromhometoroam.com/category/travels/middle-east/israel/

You might want to leave israel unless you can obtain reasonable lodging. You can go to Eilat and cross into Jordan there, spend one night in Petra, and 1 night in Wadi Rum (you certainly do not need 2 days there), and then go to Amman, visit Jerash, and then leave for Istanbul.
Whatever you decide, have a great trip.

You need far more than one morning in Petra. I had two full days (three nights) and could easily have used more. Not that much to see in Amman, but Madaba and Jerash are definitely worth visiting.

For my Jordan trip start here:
http://mytimetotravel.wordpress.com/2010/01/12/mosaics-in-madaba/

For photos of Petra: http://kwilhelm.smugmug.com/Travel/Caucasus-and-Middle-East-2009/Petra