Let me say up front that I am not, nor have I ever been—in this life, anyway—an initiate of the Mysteries of Mithras. What follows, therefore, is entirely conjectural.
I am, nevertheless, an experienced New Pagan ritualist who has been through a few ritual initiations himself.
In the absence of evidence, scholars of Mithraism have mostly refrained from attempting to recreate the ancient rites of initiation.
I, however, am not so constrained.
Here's what we know.
Ancient Mithraism was an initiatory path consisting of seven grades—degrees, one could say—each of which corresponded to one of the planets of the Ptolemaic universe.
Given, then, what we know about Mithraea—surprisingly uniform across the Roman world—with their church-like layout—narthex (entryway), central aisle, and altar-in-the-front—this is how I would do it.
The Rites of Mithras
Initiation 1°-7°
Opening Rites