Two characteristics of The Prime of Miss Jean Brodie as a postmodern novel are how the characters, especially Miss Brodie, are constructed and the author's moral concern with how much any one person can be known by another and, by extension, how much an author can truly know his/er characters. Postmodernism in literature embraces fragmentation and discontinuity and rejects modernism's belief that art can fill the void created by what became fragmented in the modern world. Postmodernism relishes that void and claims that the truth is that art cannot fill it.
Therefore, characters in postmodern literature are constructed in accord with fragmentation and discontinuity. Specifically, Muriel Spark never lets the reader into the inner life of Miss Jean Brodie. For Miss Brodie's pupils and for the reader, Miss Brodie is always the public Miss Jean Brodie, never the private woman. This is why a major portion of description of Miss Brodie comes through her mottoes, such as, "I am in my prime." Miss Brodie's life has discontinuity--there is no continuous flow from the public school teacher to the private individual with dreams and leisure and upsets of life--and it is fragmented; she is represented as a collection of random sayings that imply a unified person rather than reveal a unified person.
A more complex question is Muriel Spark's concern with the moral question about the relationship between knowing an individual or character and controlling the individual or character. The assumption is that if I know you fully--or the archetype that you (or a character) represent--then I can control your destiny because I know what will be best for you. The situation that Spark's wraps this question in is the school room, where we all follow a regime that is based on the assumption of full knowledge and appropriate control of our futures. But what if full knowledge is not possible? What if knowledge of anyone or any character is only fragmentary, like our knowledge of Miss Jean Brodie? Then what? How can attempts at control be justified? How can any voids of modern life be filled with something external like rules for order or art?
For more information on postmodern novels, see Mary Klages document, The University of Colorado, Boulder. For more about Muriel Spark's postmodern writing, see How Fiction Works by James Wood.