It is entirely possible to do exactly what you suggest, BUT please note that it will take about 4 hours of driving to get from Jerusalem to the Rabin Terminal near Eilat! And the fastest, easiest way to go is Route 1 from Jerusalem to the Dead Sea and then on Route 90 South, past the Dead Sea and straight down the Arava through the Negev Desert to Eilat. Be sure to stop along the way for refreshment and take water and fruit with you - it's going to be HOT and DRY! There's a nice area just south of the Dead Sea called Hatzeva which has an Aroma Cafe, a small shop selling snacks and pastries, etc., a Burger Ranch, clean bathrooms and more.
I suggest that a better and more comfortable way is to go down by Egged Bus #444 from Jerusalem (a cost of 65 shekels per person) and spend the night in Eilat. The next morning you can take a taxi to the crossing and continue as you said, returning in the evening to your hotel in Eilat and then catching the Egged #444 back to Jerusalem the next day.
If you would like a private, guided full-day tour of Petra, on your own or with a small group, I suggest you contact Erez at Desert Eco Tours - erez@desertecotours.com - as he can work out exactly the right sort of program for you and even help you book a hotel in Eilat.
The only difficulty I foresee is that next week 19-26 April) is Passover, and I expect that hotels and even youth hostels, etc. in Eilat and nearby are quite booked up! Still, it could be worth a try...
Have a great time, whatever you decide to do!
suzanne
suzanne pomeranz
SuzTours: Every Day an Adventure!
jerusalem, israel
suztours@gmail.comhttp://suztours.blogspot.com