What type of secondary school was attended by Adolf Hitler, and does it prepare for university?

According to William Shirer's "Rise and Fall of the Third Reich," (and I believe this was confirmed in Ron Rosenbaum's "Explaining Hitler") Hitler was initially sent to a "high school" (probably a "gymnasium") in Linz, Austria, which actually meant a school that began with the 6th grade -- he was 11. He did poorly there and was forced to transfer to the "state high school" at Steyr. He did not receive a diploma and could not have gone on to university. (See Shirer, pp. 28-35)

He could still have entered art or architectural studies at the Vienna Academy of Fine Arts - he was actually told that he had some talent for architectural design but not for painting -- but he chose not to do so.

Hitler did not attend a "gymnasium" in Linz, it was a "realschule".

The young Hitler was a good student in elementary school. But in the sixth grade, his first year of high school (Realschule) in Linz he failed and had to repeat the grade. His teachers said that he had "no desire to work." One of Hitler's fellow pupils in the Realschule was Ludwig Wittgenstein, one of the great philosophers of the 20th century. A book by Kimberley Cornish suggests that conflict between Hitler and some Jewish students, including Wittgenstein, was a critical moment in Hitler's formation as an anti-Semite. Hitler claimed his educational slump was a rebellion against his father, who wanted the boy to follow him in a career as a customs official; Hitler wanted to become a painter instead.

https://adolf--hitler.blogspot.com/2008/07/cruel-dictator-but-great-speaker.html

In most states of Germany, pupils start at a Realschule at the age of eleven or twelve and typically finish school at the age of 16, when they are expected to take on an apprenticeship. In some states, Realschulen have recently been replaced by Oberschulen or Sekundarschulen.
The Realschule is ranked between Hauptschule (lowest) and Gymnasium (highest) in the secondary school system.

https://en.wikipedia.org/wiki/Realschule

Very helpful, author. The school in Linz that Hitler attended might have led to better things, but he didn't earn any sort of qualification there or at the school in Steyr.

Shirer does claim that the Vienna School of Architecture might have accepted him: "It was still open for him despite his lack of a high school diploma -- young men who showed "special talent" were admitted without [such a] certificate -- but so far as is known, he made no such application."