Having read some of Friedrich Nietzsche's philosophical works, as well as the acclaimed biography of Nietzsche by Curtis Cate (which I suggest all those who are interested in Nietzsche should read), I understand that Nietzsche did not view Plato and his philosophy favorably, specifically, the Platonic ideal, and the notion of otherworldly truth and knowledge. Whether or not there is any record of Nietzsche's specific opinion on the allegory, I am unaware of, but I shall still try to opine on the matter. Given his scrutiny of Plato however, I'm inclined to believe it is somewhere out there.
I conjecture that the element of the story that Nietzsche would most approve of is the particular man's escape from the cave, and his venturing out into the 'real' world, out beyond the confines of the cave. Allegorically and in a Nietzschean spirit, the primitive and suffocating cave, and its residents, could represent the mediocrity of the prevailing intellectual and moral system, which would rather continue a fixation with mere reflections of 'reality' than attempt a constructive investigation of ethics and reality. My current understanding of Nietzsche lends me to the belief that this man, who has escaped the mirages and projections of what is real, would posses the opportunity of truly escaping "the herd" that resides within the cave, a term Nietzsche usually uses to describe the prevailing morality of the masses, influenced by Christian dogma as well as ossified by historical inertia. Thus, this lone man (the primordial Übermensch perhaps?), who has escaped (by chance, if I remember the allegory correctly), has triumphed over the rest in his opportunity to grasp the world as it really is. The interesting literary and situational parallels to Zarathustra are many to be considered.
As to whether there was anything 'real' outside of the cave, I believe that Nietzsche would not deny a physical reality 'of the way things are' outside of the projections and pretenses of conventional morality, and how morality influences the worldview. After all, he believed very much so, based upon what I can gather from On the Genealogy of Morality, that science during his time was one of the factors that contributed to the abandonment of traditional knowledge and virtues, so much so that he may have feared its unintended or insidious contribution to cultural nihilism. At any rate, Nietzsche did now infamously quote "there are no facts, merely interpretations", or something to that effect, though there is a greater context to that statement within The Genealogy. Still, Nietzsche was not anti-science, and within the understanding of the allegory, the man's escape from projection and duplicity towards reality and empiricism (after all, he relied upon his senses to absorb and intake the real, outside world) would be consistent with Nietzsche's urging of 'self-overcoming', which is another related concept.