You can pre book-buy tickets on-line or as you say just show up.
All tickets will have an alotted time to begin the tour. This is to assist in crowd control.
Your ticket must be inserted in a turnstile device to allow entry. If you are there at the wrong time ( + or minus a few minutes) admission is refused.
Your pre booked ticket must be picked up 30 minutes before the alloted entry time or ticket can be voided. This limits your flexibility and conveniience.
Many think it is just more convenient to just arrive, buy your ticket and wait for your admission time. Many other things at the palace to keep you busy - the gardens- carriage collection cafe- restaurant etc.
It is best to make this visit in nice weather to allow a good walk through the gardens.
I have always advised people to arrive later in the afternoon - or early in the morning. afternoon seems best.
Tour bus groups seem to be there in hundreds mid morning as mentioned before.
open hours are 8:30 - 18:00 ( 6 PM.)
If you like operettas
The Schönbrunn Palace Theater has Die Fledermaus - a nice show in a perfect setting .
for information
http://www.musik-theater-schoenbrunn.at/
July 13- August 26.
Nice combination
a palace tour - a quick meal at the Restaurant by the Schloss- then the musical evening