Trails run from St Catherine's Monastery to the peak of Mt Sinai. They are not long but can be challenging. If they were going to try and get to the top by sunrise, they would have to get up early and hike in the dark up the steps
Information from Wikitravel:
Climbing Mount Sinai is the main objective for most visitors. There are two routes to choose from, entrances to which are rather poorly signposted, so choose carefully especially if climbing at night. Both paths lead to natural amphitheater known as Elijah's Hollow or the Seven Elders of Israel, where you'll find a teahouse for a break. From there, it's a final 750 steps (30 minutes) to reach the summit.
- Steps of Penitence. This aptly named steep path has been quite literally carved out of the rock by monks, and with 3750 steps to climb is the shorter but more challenging of the two. No mountain-climbing equipment required though, just strong thighs, and the views back down into the valley are superb even in the middle of the night. If the sky is clear the stars above can also be a spellbinding sight, as there are no major cities for miles and miles around. The route can be climbed in as little as 45 minutes, or as long as three hours, depending on your pace.
- Camel Path (Siket El Bashait). A gentler, winding, wide path that snakes its way up to the summit that can be completed by the average traveler in 2.5 hours. As you can guess from the name, you can also climb up this way on the back of camel, and the "sawdust" that appears to coat much of the length of the path is in fact dried camel byproducts from the many trips before you. Be warned that the camel will bite you if it doesn't like you, so be nice, and a 4-hour ride up a mountain may not be the easiest introduction to the notoriously swaying gait of these camels.
At the top you will find a small chapel dedicated to the Holy Trinity, constructed in 1934 on the ruins of an older 16th-century church. Better yet, if you timed your ascent right, you can see the sunrise over the parched, rocky expanse of the Sinai.
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