Except in this case, an iPhone user would likely be alerted to the presence of the AirTag, because they would be moving with it from the PO Box back to their home.
Now, whether they would get the alert in time to avoid divulging their home address is another matter entirely, but it's almost guaranteed that they would get one.
Further, if the package takes more than three days to arrive, the AirTag should start beeping as soon as it's picked up. The catch, however, is that if it's buried inside a package the person may not hear it, and it doesn't seem to keep beeping.
If the person receiving the AirTag is not an iPhone user, they wouldn't get any alerts, but they also might be harder to track. If they live in a detached home without any Apple devices, you might get their neighbourhood from other nearby iPhones, but you wouldn't get their exact home address.