For a partial answer, you can work backwards from the birth dates of prominent students. It was customary to start this kind of education after finishing high school at the age of 18: A Benjamin Strasser (1888–1955) did so in 1905.
As for the admission process, the place and time suggests to me that once the applicant was able to muster the basic requirements (high school diploma and suitable examples of artistic works) it would most likely have come down to an individual decision by the admitting faculty member: much more like in an apprenticeship system than SAT scoring, say.
UPDATE: So it turned out that there is more concrete information in Brigitte Hamann's Hitler's Vienna: A Portrait of the Tyrant as a Young Man:
Like the building, the professor's college was modeled in the spirit
of the Ring Boulevard era. In art, historical painting was still the
dominating style and genre. Female students were not admitted,
inasmuch as they were alleged to bring the students' overall level of
by virtue of their dilettantism. In short: the Academy, which, in Oskar Kokoschka's words, people attended "clad in velvet robe and beret" in
order "to be considered artists", was a refuge for conservatives.
("Superruffian" Kokoschka, three years Hitler's senior, attended the
Academy's modern counterpart, the School of Arts and Crafts.) [...]
The examination procedure then was hardly different from today: the
decision whether a candidate qualified for even taking the exam is
based on the works submitted. In early 1907 Hitler was one of 112
candidates who would take the admission tests: he set out with a pile
of drawings, convinced that it would be child's play to pass the
examination. He passed this first part of the test and was admitted to
the drawing exam [...]
The academy's drawing exam was administered under
supervision on October 1 and 2, 1907, in two separate groups, lasting
three hours each in the morning and the afternoon. Eight "composition
tasks" had to be carried out from one group of themes each, such as
"(1) Expulsion from Paradise (2) Hunting (3) Spring (4) Construction
Workers (5) Death (6) Rain." This time Hitler's works did not meet the
requirements. The "Classification List of the General Painting School
1905-1911" [sic!] contains the entry: "Adolf Hitler, b. in Braunau on
the Inn, Upper Austria on 20 April 1889, German, Catholic senior
official of the Dual Monarchy (father), few heads," and the result:
"Drawing exam unsatisfactory." Of 113 candidates who showed up, only
28 were accepted into the painting school, which appropriately equals
today's admission percentage [...]
Speculations tracing Hitler's
anti-Semitism back to his rejection by Jewish Academy professors are
entirely unfounded: not one of the responsible men during the
examination was Jewish.
Hamann cites her original sources (as one would expect), and she also names the key faculty members involved (in a fragment I did not yet cite).
