So I understand most of Mozart's Italian-language operas were originally performed in German-speaking places like Vienna and Prague. Would the audiences of these premieres have understood the goings on in the libretto? Indeed, would Mozart himself have been fluent in Italian, or relied on his librettist for translations? On a related note, how common was fluency in several languages among Western European upper-middle class and nobility at the time?
The answer to your questions is a qualified yes.
Mozart went on three Italian journeys in his youth where he picked up Italian motifs and the language. One of his 1770 letters to his sister goes "now the German dope ends, and the little Italian dope begins," and he proceeds to congratulate her in Italian for being more fluent than he thought.
As for the audience, their response to the Italian in operas was more nuanced and variegated. This was partly because opera in the eighteenth century was less centered around the music than it would later become. A 1783 essay by Stefano Atreaga classified opera goers into several categories. The most common opera goer, the genti de mondo, were the audience members who were there as a social occasion and were there to be seen. Atreaga also classified some as politicians who used opera as a means to hobnob with the ruling elite. There were also philosophers, who understood the meaning of the music and how it reflected national character and the men of good taste who appreciated the aesthetics of music. His final classification, and the one most pertinent to the OP's question, were the pendants pedants who are not emotionally involved and check the libretto for factual or historical inaccuracies and completely miss the point of the music.
Arteaga's scorn for the pendants pedants underscores that baroque music emphasized affekt or emotional themes in music. The goal of composers and performers was less to get the audience to understand the plot and instead feel it. Of course, artistic aspirations are different from reality. Austrian and German theaters would often sell librettos with dual German-Italian translations to their audience. Librettos also had to be produced for state authorities so that they could pass the censor, so the marketing of an existing product had a certain logic to it.
The multilingual nature of the upper classes (there really was not much of a middle class as we would define it during the eighteenth century) and nobility was something well attested to. The standard education involved tutoring in foreign languages as part of a general process of culture. That said, proficiency in languages could be quite variable. Private tutoring followed no real set pedagogical practice and was dependent upon the quality of the tutor. Marie Antoinette's French at her arrival at court was notoriously bad and accented.
Sources
Buelow, George J. A History of Baroque Music. Bloomington, IN [u.a.]: Indiana Univ. Press, 2004.
DelDonna, Anthony, and Pierpaolo Polzonetti. The Cambridge Companion to Eighteenth-Century Opera. Cambridge, U.K.: Cambridge University Press, 2009.
Mozart, Wolfgang Amadeus, and Robert Spaethling. Mozart's Letters, Mozart's Life: Selected Letters. New York: Norton, 2000.