Can I drive my rental car across the border?

No. Rental companies prohibit taking cars across the King Hussein Bridge. You must drop your car off in Amman and pick up a new one in Israel, or use taxis.

Is it worth doing this in one day?

It is physically possible, but you will spend 4-6 hours in transit. If you are short on time, it works; if you want a relaxed trip, stay overnight in Jerusalem.

How to Cross the Border Between Jordan and Israel for a Day Trip

Crossing the Jordan-Israel border for a day trip is possible via the King Hussein (Allenby) Bridge, but it requires advance planning because you cannot drive a rental car across. You must use a combination of official shuttle buses and taxis, and you must check the border status 24 hours in advance, as closures are common.

Check the border status. Before booking anything, check the official Israel Airports Authority website for the King Hussein Bridge status. It is the only crossing near Amman/Jerusalem; the northern and southern crossings (Sheikh Hussein and Wadi Araba) are too far for a functional day trip.
Arrange ground transport to the border. Book a private taxi to take you from your hotel in Amman to the King Hussein Bridge terminal (about 1 hour). Do not expect to find a reliable public bus at 6:00 AM.
Navigate the terminal process. Pay the departure tax (10 JOD in Jordan). Board the JETT shuttle bus to cross the 'no-man's land' (about 10 JOD). You will then go through security and passport control on the Israeli side.
Secure onward transport. Once you clear the Israeli terminal, take a sherut (shared taxi) or a private taxi to Jerusalem. The ride takes about 45-60 minutes.
Return before cutoff. The border usually closes to incoming traffic early (often 2:00 PM or 3:00 PM). You must be back at the Israeli terminal well before the final closing time to ensure you aren't stranded.

Can I drive my rental car across the border?: No. Rental companies prohibit taking cars across the King Hussein Bridge. You must drop your car off in Amman and pick up a new one in Israel, or use taxis.
Is it worth doing this in one day?: It is physically possible, but you will spend 4-6 hours in transit. If you are short on time, it works; if you want a relaxed trip, stay overnight in Jerusalem.