While at that point in time, most people already used tempered tunings (though not always quite the same equal temperament as now), the reference tone a' was still subject to change. It is actually fairly difficult to find out how high or low it actually was, and I'm not quite sure about any exact numbers on Mozart. However, there is some evidence (a tuning fork at least, I believe) that eg Händel's a' was around 420Hz - meaning A LOT lower than today (generally assumed ~440Hz).
I've heard this is actually part of the reason why some old string instruments are starting to break down, perhaps most notably some old instruments by Amati, because they were simply not built for the high pressure going along with the high tension in the strings required by a high pitch. Baroque instruments - as still widely used in Mozart's time - might not take well to that pressure, due to the different angle of their necks.
It stands to reason (and is taught at the music university in Vienna), that instruments still were tuned notably lower than today at Mozart's time. Hence, not as dreadfully high to sing as today. You might want to compare some recordings or Mozart played in the mid-20th century to some recordings by, say, Harnoncourt or any other orchestra specialising in old music. The first couple bars of the same piece should be enough. It's not always the case, but often times you will hear a distinct difference in pitch.
The phenomenon of rising pitch I have sometimes heard referred to as pitch inflation. The Vienna Philharmonics' a' is, as far as I know, around 445Hz already, the most common around here (Vienna) in various orchestras is 443Hz, according to what I have experienced.
To sum it up, the tuning in itself might have been the same (or fairly similar, at least), but since the reference tone a' has risen notably over the last few centuries, the f''' is still very likely to have been notably lower as well.