Unfortunately St Catherine’s closes for various religious days and, to complicate matters further, is still operating on the Julian, not the Gregorian calendar! I know of no way to check in advance whether it will be open on any particular day or not.
So, on my second visit we could not get in (luckily I had already visited the “Burning bush” and the library). There is still Mount Sinai to climb however! It provides some fine views down onto the monastery and a wide panorama at the top. It will take about 3hrs. It is “traditional” to do it at night (when you can make it more of a “pilgrimage”) for the sunrise but if (as we are not) you are not a great lover of sunrises it is a reasonable climb in winter before the sun rises too high.