1. Try doing a hard reset:
Press and hold both the Home button and the Sleep/wake button simultaneously for about 15 seconds and release when the Apple logo appears.
2. If that doesn't solve the issue, backup the device and restore it as new.
In that case, use: http://support.apple.com/kb/HT4137
NB: set up as a NEW device because a software issue will be in the backup. If the issue persists after restoring as new, you should offer the device for service.
If you get specific error messages/codes in iTunes when updating/restoring, check this: http://support.apple.com/kb/ts3694 for causes and solutions
I also have this problem -- and it Is NOT the first time this has happened during an iOS upgrade! I missed several important phone calls yesterday because the sounds don't work after this upgrade! I did a hard-reboot as suggested, and it still doesn't work. Grrr!! Now I have to go through a restore to get my sounds back AGAIN! This truely ***!! Why does this happen? And can it be avoided in the future? Makes me not want to ever upgrade my iPhone ever again!