Most has been mentioned about the monastery itself, so I won’t go into details. According to their website, the monastery is closed on Friday and Sunday, so I picked Saturday to go there. Further the museum and church closed at 11:30 that day, so better early if you want to visit them. We also skipped visiting the mountain. If you are not religious, there is in my opinion little incentive to climb such a mountain.
I went to the monastery from Sharm El Sheikh by rental car and apart from the usual police and military blockades this was no problem at all. No special permit needed and no police escort. Streets are great and empty the whole way. It is very easy to find as there are not many streets and signage is good. A clear advantage for the crazy ones like me is that you can visit 3 TWHS on the day trip to the area. We stopped in Dahab in the morning and took the longer route (had to convince the police that I want to take a detour) on the way back, with stops at Wadi Feiran and Raitho monastery. All 3 TWHS are from 1994 and will never make it to the list. So don’t bother if you are not that serious.