It's possible to see the moon at noon (probably almost setting), and then see it rise at 9:30 PM.
We are all familiar with the fact that the length of sunlight varies during the year. For mid-northern latitudes, the sun is up for about 15 hours in June (and down for 9), and is up for about 9 hours in December (and down for 15).
If the moon is near setting at noon and rises at 9:30 PM, that means that it was down for about 9 hours. That is perfectly possible if both of the following conditions hold. (I'll assume that you're in the northern hemisphere.)
1) You're either at mid-northern latitudes or even farther north. (If you were at the equator, the moon would always be up for about 12 hours and down for 12 hours; so you could never see it rise and set at the times you described.)
2) The moon is at a very positive declination. (That means that for observers at the latitudes of the US or Europe, the moon will be above the horizon for more hours than it is below the horizon.)
Let me work on this for a few minutes and I'll come up with a concrete example.
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OK, here's an example. Suppose you're at latitude 45N, longitude 75W (near Ottawa, Ontario, Canada), and look at the moon this summer on September 10. The moon will be just before last quarter. It will set that day at 1:18 PM EDT, and rise at 9:58 PM EDT. (This example works even better as you move east in the world on that day. You can find a place where it sets about 12:50 PM and rises about 9:30 PM, matching the times that you gave.)