Mozart himself didn't write any of the texts (libretti) of his operas. He "only" wrote the music after a given text. And more than 70% of his operas are in Italian language (one, Apollo et Hyacinthus, KV 38, is even written in Latin).
There are only four Mozart-operas with German libretti (five, if you also count the fragment Zaide, which in fact is a unfinished first version of Die Entführung aus dem Serail). Among those four German Mozart operas only two are well known:
- Die Entführung aus dem Serail (KV 384, premiere 1782), libretto written by Johann Gottlieb Stephanie d.J.
- Die Zauberflöte (KV 620, premiere 1791, two month before Mozart died), libretto written by Emanuel Schikaneder
The two other German Mozart operas are »Bastien und Bastienne« (1768, KV 50) and »Der Schauspieldirektor« (1786, KV 486). But both are short (less then one hour each) and not often played. And again: Mozart did not write any of the words, he only wrote the music.
So when you really asks for Mozart's German, then you will not find any trace of it in his operas. You have to read his letters. You can find a complete list of all his letters and notes here: http://dme.mozarteum.at/DME/briefe/doclist.php
The German of »Magic Flute«
Emanuel Schikaneder wrote the libretto of this opera, and you can find the complete original text here: http://www.e-german-up.net/IMG/pdf/original_libretto.pdf
The German used in this opera is not typical for Vienna of the late 18th century, nor is it typical for the German of any other place or any other time. It is an poetic and artistic language, and the libretto contains lots of hidden reference to the freemasons (Schikaneder was a freemason as well as Mozart).
This text is more than 200 years old and was written at a time where there was no standardized orthography (German orthography was first time standardized in 1876, 85 years after the Magic Flute was written).
When listening to the opera, it is even hard to understand the words of the sung parts if you are a German native speaker who knows the opera very well (like me). A German native speaker can read the libretto and will easily be able to understand the words, but It's hard to get all those freemason hints. I guess some knowledge that would be needed to understand the full meaning of this opera got lost over the centuries.
In a good English translation of the libretto you will find everything that is needed to understand the opera on the same level that is available for a German native speaker. So there is no need (except for academic research) to learn the German used in this opera.
Addendum (not part of the answer):
Although I am Austrian (born in Graz, living in Vienna), I have to correct a very common mistake. You wrote, that Mozart »was Austrian by birth«. This is wrong:
Wolfgangus Theophilus Mozart (who later changed his name to Wolfgang Amadeus Mozart) was neither German nor Austrian. He was born in the year 1756 in the city of Salzburg, which was the capital city of the Archbishopric of Salzburg, which was a sovereign nation until 1805. Salzburg did not belong to any other nation. It was a state on the same level as France, Bavaria or Austria, just smaller.
In particular, during Mozart's complete lifetime (1756-1791) Salzburg was not part of the Austrian Empire, nor was it part of any other nation. In 1805 Salzburg was annexed to the Austrian Empire, and since then it was part of Austria (and still is). Although Mozart lived in Vienna for the last 10 years of his short life (he died at the age of 35), he never became an Austrian citizen. From his birth to his death he always was a Salzburgish citizen.
But since Salzburg belongs to Austria nowadays, Mozart often is called an Austrian composer, and Austria (especially Austrian tourist branch) is very proud of that. But to say, that he »was Austrian by birth« is wrong.