Just made the trip to Jerusalem two days ago! (Of course, the situation changes all the time, so this could well be irrelevant by the time you get here)
I took a private taxi from Amman to King Hussein Bridge (25JD). I hear tell of buses, but I haven't seen one yet. Maybe someone can offer up some info on these mystical beasts? Service taxis are an option.
If the taxi doesn't drop you off exactly where you need to be, just ask around and there are plenty of people who will direct you.
There's a building marked "Departures" but you'll figure out pretty quickly that it's closed. You need to go to the one marked "Arrivals." On through the metal detector and up to the window. 10 JD exit tax and they'll take your passport (This always makes me nervous, but you'll get it back when you get on the bus).
I want to tell you not to worry about the whole "Israel stamped passport" issue, but I'd hate to tell you that and be wrong about it, so I'll just tell you what happened to me and the 2 other people I was traveling with. They have a separate sheet of paper that they stamp in place of your passport. If you're really concerned about it make sure you say something like "No stamp" when you hand over your passport. What seems to be unavoidable is the sticker they will put on the outside of your passport. I've read that border control agents in Arab countries look for the residue from this sticker as an indicator that you have visited Israel, but there are also instructions online on how to remove this sticker without leaving a trace, so we'll see.
We waited about an hour outside under the covered area that has a "VIP business" sign on it. There are buses that pull in and out of there all the time and I was growing a little concerned about which one was ours, but after asking around someone assured me it would show up eventually. If there are a lot of people waiting, you'll probably know when it arrives because they'll all get up and crowd in, but it there are few, someone will most likely alert you when it arrives.
The bus was 7JD (for a 3 minute ride???) and 1.5JD for each luggage item (carry-ons are excepted). We got our passports back before the bus left, so hopefully you will as well.
When you arrive at the terminal on the other side (after one or two checkpoints) You'll go through one passport check where they look at your passport and send you through the clear plastic drape things that remind me of ET. Then another checkpoint that is pretty similar to airport security. Bag/jacket in basket, no phone in pocket, no belt, no keys or change. Keep your passport on you because immediately upon going through the metal detector they will perform another passport check.
After gathering your belongings continue through the terminal until you get to yet another passport check where they will ask you a few questions (Do you know anyone in Palestine? Why are you going? How long will you stay?) and you will receive your visa. These guys were in good spirits when I went through, but it also was a fairly quiet day.
Continue on through 2 more passport checks (seriously, Israel?) and you're pretty much there. There's a money changer just before the door that leads outside, and an ATM somewhere. You'll need shekels for the ride to Jerusalem, so I guess that's kind of your only option.
The Al-Nijmeh is a 10-seater bus/van that costs 42 NIS a head, and it will take you as far as the Golden Walls Hotel (a few hundred meters from Damascus Gate). Tickets for the Al-Nijmeh can be purchased from the kiosk that is to the left as you are leaving the terminal. You can take a private taxi, but I stopped listening to the driver as soon as he said something about 300 shekels.
On the way to Jerusalem we were stopped at a checkpoint and asked to show our passports with visas. The driver will most likely ask you where you want to go. Ours was quite accommodating.
Getting back to Jordan went well. If you're curious about that, let me know and I'll post that information as well.
We stayed at the New Palm Hostel and Hotel. Mixed reviews online. Loud, a little dirty, but it's cheap, is in a prime location and has a free (albeit mediocre) breakfast and WiFi (also mediocre). I would recommend it to my friends but not my mom. If you stay there, try to get a room in the hotel wing.
Ahlan oo sahlan!