In today's phone directory of Austria (www.etb.at
), the name is spelt Poschenreiter, Poschenreither, and Poschenreithner. However, it does not appear in Graz or Styria. Nevertheless, it is of course possible that your ancestor was born in Graz.
If Franz was born in Graz, there are a lot of resources available for further search. First of all, LDS (the Mormons) have films of the male birth record duplicates of Graz which were made for purposes of the military draft. Since you have an exact birthdate, you could search the related film and thus obtain data on the parents, where they lived in Graz, job of the father, and more. The Mormons have also filmed the residency records of Graz. The films span the residency books (Meldebücher) written from 1876 till 1892, and the residency forms (Meldezettel) covering 1892 till 1925. The City Archive of Graz also has the census records of the 1880, 1890, 1900 and 1910 censuses, all with very detailed information, especially on the birth places of the counted inhabitants.
If Franz Poschenreiter was really born in Graz, one could find out much more about him with certainty.