I travelled as part of a group from a cruise ship, so we had a number of sites in a day's bundle all prepaid. A visit through a tour company would include the Old City at a cost of about $20. Otherwise you could simply drive there yourself as there are no toll gates.
The Mount of Olives is not just a building or so, it is an area which is free to access.
There is no charge to visit the mountain itself. Most of the Holy Sites are free although the clergy who run them would be grateful for a donation. There are donation boxes at the rear of all churches. A few sites such as the Pater Noster (Lord's Prayer) charge a small entrance fee.
As someone mentioned is a public area.you only pay at the place where Jesus really ascended into heaven,but it is voluntarily because it's on a private property.
We were on a tour and this was one place we did not have a fee for. It is close to the Garden of Gethseme (donations accepted) and the vacant tomb. (No charge).
It is a public area, no fee at all.
Mount of Olives is free for public visits. Your car or coach bus can not park near the entrance gate, so you need to walk on the nice pedestrian line. Spare some time to visit also The Church of Nations, the place (table-like stone) where Jesus prayed in the garden, in the same location as Mount of Olives.
No fee, better to go by car and walk down. For walking up to the Mount of Olives is a steep climb, doable but still steep.
There is a hand rail on the wall to hold on to.
You can walk down from the Mount of Olives to Dominus Flevit Church, then Garden of Gethsemane, then Tomb of the Virgin, then St. Anne's Church in the Old City.
Take water and or snacks.
Once inside the old city you can do the rest of the Holy sites.
Google and make sure you click on "walking directions."
Enjoy, it is beautiful.
Indeed, as others beat me to the answer, it is completely free, is open to the public, is easily accessed, is adjacent to the Church of the Nations, affords great views over the city and is, on its own, a very peaceful and spiritual place. I will add that the chap with the camel is easily avoided if you visit for spiritual rather than commercial reasons!
There is no fee. FYI - the is usually a Palestinian gentleman with a camel on the side of the road where the tourist groups stop. You can have your picture taken with your camera while up on the camel for a cost of $5. Suggestion - have a friend take several pictures of you on the camel so that you are not dependent on the one or two pictures that are taken by the camel handler's representative. It's an inexpensive opportunity of a lifetime to have your picture taken with the old city and temple mount in the background.
there are no fees. You can get there by car or on foot.
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