The cheapest way I know is to get a bus from Nazareth all the way to Amman using a company called Nazarene Tours. The bus departs almost daily at 8AM and will cost around $20. You will be dropped off around Amman University. A public bus from Tel Aviv to Nazareth will cost you less than $10.
Another company called Mazada Tours has a direct bus from Tel Aviv ...
The current Foreign Office travel advice for Jordan warns against travel close to the Syrian border, for obvious reasons. Travel to other areas of Jordan is fine. You should familiarise yourself with the information they provide about safety etc. although most of it is common sense.
It also says that over sixty thousand British nationals visited Jordan in ...
If you are not traveling alone you can use a Taxi, which could run you by my calculation about $250pp.
If you want it even cheaper you can do a more complicated route of Tel Aviv - Eilat (#393, #394), Eilat - Aqaba and finally Aqaba - Amman.
Or you can do a straight trip as described on True Nomads, which would all run a lot less then $400 for a flight.
You never have to right to fly on a certain flight, even if you have a boarding pass.
The airline can always 'bump' you, for whatever reason they chose, or without giving a reason.
All you have is the right for compensation. The details depends on the country, and on the airline / the fine print in your ticket. At minimum, you'll get your money back; often ...
From the information you have supplied here, you were wrongly denied boarding. Your U.S. visa was sufficient to make the transit:
Information as of 23AUG15 / 0855 UTC
National Syria (SY) /Residence Jordan (JO)
Embarkation Jordan (JO) /Transit Germany (DE)
Destination USA (US)
ALSO CHECK DESTINATION INFORMATION BELOW