From Tel Aviv Central Station, board the Egged bus n°843, on platform 5 to Beit She’an Terminal Station (37,5 NIS). You can find the time schedule here.
I would recommend to take the 9:50 AM bus, as you’ll then be able to board the 12:30 bus 16 at the Beit She’an terminal station, that will bring you closer to the border (there is no stop at the border but you can try to ask the bus driver, they apparently usually agree to drop you at the border entrance).
There is no bus between 12:30 and 16:30, so the only option you’ll have if you don’t want to wait or walk the 7k will be to take a taxi. The ride to the border takes less than 10 minutes and will cost you 50 NIS. You can try to negotiate but I wasn’t successful as the taxi drivers know you have no other option, so try to find people to share the ride with.
Crossing the border is pretty straightforward, you walk through the Israeli customs, pay the exit fee (105 NIS), then wait for a shuttle that will bring you to the Jordan customs (5NIS). There, you’ll get your visa (40JD) and enter Jordan. Prepare cash as there is no ATM and they don’t accept credit or debit cards, but you can exchange to JD or NIS at a fair rate.
There is no public transportation at the border, so the cheapest way to get to Amman, is to take a taxi to the city of Irbid, that costs around 20JD, and from there finding a bus to the capital.
Another option is to have your Couchsurfing host pick you up at the border, in exchange of a compensation to cover for the fuel and hassle.