Was Plato a mystic? That's hard to say. But whether he himself was, aspects of his thought certainly contribute to later mystical thought. In fact, it is hard to imagine mysticism without Plato. Why?
1. He has a distinction between appearance and reality, which becomes central to mysticism.
2. He has a distinction between the soul and the body, and believes that the soul is eternal. See Socrates' speech in the Phaedrus for this.
3. He does not distinguish between intellectual knowledge and emotional attachment. His work on Beauty, for instance, establishes that our attraction to beauty is ultimately an attraction to the intellectual forms. So, that which truly motivates us is also that which is worthy of contemplation, and can be truly known.
4. He has a story of ascent, particularly in the allegory of the cave.