Refugium mud is completely unnecessary. In fact the specialized mud that is sold in the hobby is really very similar to the dirt from your backyard. When I have grown Mangroves, and I've grown a lot, I have wanted them to assist with tank filtration. Using mud would have been counter to that purpose. Mangroves grow perfectly fine in the same Aragonite sand that we use in the main tank.
Mangroves must get a footing in a sand/rubble bed before they can start to grow. This means they need to be left undisturbed and allowed to grow some roots first, before leaves and branches will begin to grow. This can take anywhere from 2-4 months. As such, they can be added at the very first.
Cycle? Setup of a new reef aquarium does not have to go through the old time cycle of pollution that kills any higher animal that might be added to the tank during the dreaded "cycle" time. The Reefkeeping Tips thread linked below explains more about how to completely avoid pollution(cycle) in a beginning reef.
Aloha,
Mark
