Mozart was and is one of the most influential musicians and today nearly everyone knows his name and probably some of his famous compositions. During his lifetime he certainly was a well known musician at least in the royal courts of Europe. But would a common person in Europe or the Holy Roman Empire like a farmer or a tavern owner have heard Mozart's music during his lifetime?
Would they have even heard about him or did his name and music spread much later?
Tickets to the première of the Magic Flute in 1791 were available at the theater's normal prices, which were, when it opened four years earlier, as low as 7 Kreuzer for the gallery. A box seating eight people was 5 florins or 37.5 Kreuzer per person.
In a letter of 1777, Mozart writes of the price of copying a sheet of music being between 6 and 24 Kreuzer. I couldn't find data for the cost of everyday items in 1790s Vienna, but I suppose that it isn't unreasonable to think that a tavern owner could afford such prices. The theater and the piece Schikaneder commissioned for it were, after all, directed toward a different market from that of the Italian opera of the nobility.
Mozart lived in Modern Europe (died 1791, soon after the French Revolution); not in medieval, obscure times. Mozart's music was sold (print music sheets), adapted, re-arranged, played by memory, played at home, on the streets, on theaters, etc., during his lifetime enough so that most people in Central European countries would have heard it. Most people would have listened to enough Mozart to recognize his personal style, top melodies, etc.
Music prints were generally available (see: earliest Mozart editions) and although expensive, people copied them themselves at home very cheaply (also, cheap copies were sold). Playing the violin was quite popular, and playing the piano was to become very popular very soon (think 1810). Small string orchestras were also very common in all the German-speaking area to play symphonies and such.
Mozart was a famous child prodigy touring courts, and then, later in life he become "a normal composer". Although famous (think like a Kardashian), he often struggled financially, just like any musician, because at the time, music was not yet an industry and was not perceived as something "useful" people paid for.
What is important to add to the other answers is that Mozart music became popular before the emergence of modern democratizing mass media (like radio and television). This has two somewhat contradictory consequences:
As the other answers have pointed out, the access to mass entertainment was somewhat popularized from XVII-th to XIX-th century due to the expansion of small bourgeoisie (equivalent of modern class.) However, I would think that lower classes also had access to theaters - in the same way as more than two centuries earlier they had already had access to Shakespear's Globe theater.
Those living in the countryside might have heard of Mozart via wandering musicians. Pushkin in Mozart and Salieri, published in 1832, describes a (fictional) scene where Mozart is amused by a blind street violinist rendition of Mozart's music (translation borrowed from here):
Mozart
Just now. I had
Something to show you; I was on my way,
But passing by an inn, all of a sudden
I heard a violin... My friend Salieri,
In your whole life you haven't heard anything
So funny: this blind fiddler in the inn
Was playing the "voi che sapete". Wondrous!
I couldn't keep myself from bringing him
To treat you to his art.
Entrez, maestro!(Enter a blind old man with a violin.)
Some Mozart, now!
(The old man plays an aria from Don Giovanni; Mozart roars with laughter.)
