I was having some serious issues with my phone including it asking for my PW all the time, I'm sure you all can relate to that one. In addition, my iPhone would not backup to icloud either. To resolve this issue, I spent over 5 hours on the phone with various apple employees. Nobody could figure it out so eventually my claim got escalated to a senior advisor. We tried several trouble shooting options (hence the long time on the phone). I will spare you the tediousness of non-solutions and skip right to what has seemed to work.
Here is what we ended up doing:
Step 1: Save any pictures, contacts, etc. that are important (even though my phone wasn't syncing with icloud, when I plugged it in I still had access to my photos).
Step 2: Restore the iphone to factory setting (I know this sounded like a sucky option for me, but it worked and not too much was lost)
Step 3: Once my iphone was restored, I had to my iphone and go into settings – general – reset – reset all settings. From there it gave me the option to restore my iphone from a previous icloud backup (My last backup just so happened to be the day before I installed the ios6 update, I would bet that yours is probably similar).
Step 4: Once I had the iphone restored, my apps kept crashing so I deleted all of those and reinstalled them thru itunes.
I have had no issues with it since doing that about 3 weeks ago. I know this is a pain to correct and it was annoying, but given the time I spent talking to Apple employees and their general confusion with this issue, I think this is probably going to be the easiest solution. Hope this helps!