Loved seeing this place! Travelled by myself by bus to this place in the early morning... Asked at the main bus stop in Eilat which bus I should take and it was a bus that goes along the Egyptian border to Be'er Sheva (check Google maps). Ask the driver to stop at the road that leads to the Canyon. From there it's a few kilometres hike to the entrance of the Canyon. Take plenty of water with you! The Canyon itself is amazing to see. Love the colours and shapes of the rock. Reminded me a lot of the city built in the rock in Jordan called Petra. It's not very long though, but for me it was very special to see as it reminded me of Petra. Hiking back there were a couple of groups of school kids about to enter the Canyon which was nice to see, cause until that stage it had just been me.
There is an (expensive) tour that can take you to the Red Canyon by jeep, but for me I like to do things my own way and it's very possible to go here by local bus if you don't mind the extra hike to the entrance and take plenty of water with you. It was very beautiful!