How do I visit Petra from Israel?

Yes, (as of 2002) you can cross at Allenby bridge and then take a mini-bus to Petra; however, you'll want to get your visa in advance. Crossing at Eilat would probably be faster.

You will probably want to spend two nights at Petra, so that you can have a travel day, a day to visit the ruins, and then another travel day. There are many inexpensive hostels around Petra; you won't have trouble finding one. I stayed at "Hotel Valentine".

I posted some information on how to travel from Israel to Jordan in the question "Jordanian visa rules when coming from Israel".

From what I know going into to Jordan from Israel is harder than going from Israel to Jordan. If you haven't booked your trip yet I would consider flying into Jordan then making your way to Israel. If possible I would stay for 2 nights for what it's going to cost you ro get there. You can see the Wadi Rum while you are there as well. I didn't look into one day tours from Israel but I did see companies that were offering 2 or 3 days but that was from Egypt. But Egypt doesn't have the same complex relationship that Israel has with other countries in the region.

At least at present, yes, there are tour companies that do 1-day tours to Petra from Eilat. I was on such a tour about 6 weeks ago (i.e. November 2016.) I wouldn't recommend trying to do a day trip from the Allenby crossing, though, as that would be a rather long drive for a day trip, about 3.5 hours each way according to Google Maps. From the Eilat/Aqaba crossing, it was around 2 hours each way to/from Petra, as I recall. Google Maps agrees with my remembrance.

Crossing the border at Eilat was relatively trivial in both directions. Entering and exiting Israel via the Eilat crossing was much faster than exiting Israel via Ben Gurion Airport. However, judging from Jordan's website, it appears that visa-on-arrival at the Wadi Araba (Eilat/Aqaba) crossing now requires being part of a tour group that is arranged ahead of time or else obtaining a visa ahead of your travel.

Starting 1st of January 2016, entry visas will no longer be issued at the Wadi Araba Border Crossing. The only exceptions are Israeli tour groups and others carrying proper entry documentation entering Jordan for tourism purposes. Moreover, Israeli tourists and those carrying proper entry documents who have bought the 90JOD entrance ticket to the Petra Archeological Site, will be exempted from the obligatory 24 hour stay in Jordan under the following conditions:

1- Local tour operators should submit manifest lists and travel documents of the touristic group to the Tourism and Marketing Directorate of the Aqaba Special Economic Zone Authority at a minimum of 48 hour prior to the group’s arrival to the country, so that visas and clearances can be issued smoothly in accordance with laws and procedures of ASEZA.

2- Local tour operators should submit necessary guarantees that all members of the touristic group will leave together as one group through the Wadi Araba Border Crossing and not through any other point of departure.

3- All members of the touristic group included in the original submitted manifest should be present upon departure; permissions for departure will not be granted to groups in which members included in the original manifest are not present.

4- No entry visa will be issued for any other nationality through the Wadi Araba Crossing and entrance will only be granted to those who have obtained an entry visa to Jordan through Jordanian embassies abroad prior to their arrival at the border crossing.

If you want to do a multi-day trip to Petra, there are several hotels within a short walking distance of the entrance to Petra. In case you're considering this, according to the same Jordanian link above, an additional benefit of spending at least two consecutive nights within Jordan is that the visa fee is waived for nationals of 'non-restricted' countries, regardless of whether traveling individually or as a group.