Middle East - Visa for US Passport Holder for Jordan

I've seen that a US passport holder requires a visa to enter Jordan, that costs roughly $30, which is valid for one entry, and valid for one month. (And obviously higher costs for longer than one month and/or multiple entries.) I've also seen that one will be provided automatically when landing in AMM. Does that mean that upon landing the fee is waived or simply I can buy a visa there without applying it for first (meaning I don't need the visa when I board the AMM-bound flight.)
What about land crossings into Jordan? I know of at least 2 crossings...one by the Allenby Bridge and one in Eilat (is it actually in Eilat?). Can I buy the visa from either of those crossings?

If I need to apply, how far in advance should I apply?

Thanks!

You can buy the visa at the airport once you arrive in Amman. You do not need to apply for the visa before hand. The fee is never waived.

You cannot buy a visa at the Allenby Bridge crossing. You will need to have a visa issued before hand from from a Jordanian Embassy or Consulate. I am not quite sure how much time before you need to apply but to be on the safe side I would say one week (or it could take just a day.) If you have a valid multiple entry visa from a previous visit then that would also work at the Allenby Bridge.

All other crossings (including Eilat/Aqaba) do issue visas at the border.