Is this something that can easily be arranged with a tourist agency or hotel? Next March. 2 women.
We were looking to do the same thing - there are tour companies that will pick you up from Eilat in Israel in the evening, to do the climb of Mt Sinai starting at 2am, and visit St Catherine's monastery in the morning, and return to Eilat. You could arrange this and then make your own arrangements to get to/from Eilat.
There are also apparently two ferries (fast/slow) which run to the Sinai peninsula from Aqaba, but it sounded complicated/time consuming. I wasn't able to find a tour operator which would offer the tour from Aqaba, as unless you go by boat it would be two border crossings.
In the end, we decided against going, as we couldn't get any travel insurance cover the trip (most countries recommend against travel to the Sinai peninsula), and felt we had more than enough other things to fill our time.
That's a long way, there and back and climb! It really isn't possible in one day, even two days is pushing it.
Logistics are easy enough; leave out the ferries, unless you want to avoid an Israeli passport stamp, take a taxi to the Israeli border as early in the morning as you can, cross the border and take another taxi to the Egyptian border, buses and service taxis leave for Sharm and St Catherine from the Taibeh bus station which is about 100 meters from the border and take about 4 to 5 hours. At St Catherine's bus stop, it's easy enough to find a guide, try to do the overnight climb to see the sunrise from the summit!