This was very interesting. We made a last minute reservation (+972-2-6271333) on our way into Jerusalem for that night. Note that it is often difficult to get a reservation. Try to book way in advanced. If there is no availability, continue to try up until that day. If you are very motivated, also try to show up since space seems to often magically appear at the last minute. If possible, try to do a day tour. During the day, you exit out the far side into the Muslim quarter. At night, you need to double back the whole route which means a lot of squeezing by people (both from the prior tours and when you need to exit). The tour was very interesting though and certainly worth doing.
Israel Jerusalem Travel Western Wall