The Scripture records that Aaron was buried on Mount Hor (Numbers 20:25-29). We wrote about this here. A few interesting questions have been left in a comment. I am away from home for the week and do not have access to all the materials I might check, but here is a brief response.
1. How long would it take to walk unaided up Mt. Hor? Are there clearly marked-out historical trails?
Hikers make the trek. My recollection is that one should allow between a half and a full day for the trip up and down. A guide book such as Lonely Planet would surely answer this question. Perhaps some reader has made the climb and will tell us.
2. Do you know anything about the history of the Muslim shrine at the summit? I’m sure there have been quite a bit of academic research into this over the years, but I’m hoping for a simple answer about how reliable is this tradition about the exact location of Mt. Hor?
Not really. There are many “traditional” places identified by Jews, Christian, and Muslims, but a large number of them are without any historical foundation. Todd Bolen has this comment, along with a nice photo, at Bible Places.
In Bedouin tradition, Jebel Haroun is Mt. Hor where Moses’ brother Aaron was buried. Most scholars reject this, locating Mt. Hor near Kadesh-barnea to the west.
I trust this will be helpful.
Here is a photo of the beginning of the Siq at Petra. Almost everyone likes to show the photos of the narrow part. It is beautiful, even from the beginning.