A friendly warning

It's an excerpt of the recording of Also sprach Zarathustra used in the film 2001: A Space Odyssey. It was made in 1959 by Herbert von Karajan and the Vienna Philharmonic. It is most definitely not under a CC license.

But do to it's incredible shortness, would it not fall safe under "Fair Use"? Well, as safe as something can be under FU...

It probably is fair use, yes. It's less than twenty seconds long; it isn't of very high quality; it could be construed as being used in a parody of its source; it isn't being sold; etc. However, it isn't credited.

Fair use cases can get pretty far into gray areas, though I'd say the use of this recording is defensible.

Also, since this is a public domain piece of music, it probably wouldn't be very hard at all to find a CC-licensed recording. (If a longer recording is to be used.)

So would there be much of a problem of uploading it to the Archive when it's finished?

I imagine there wouldn't be much problem at all, but I don't know what exactly the final submission rules are to be.