How can I restore my iphone 4s to a previous backup I can see in the backup folder?

Question

Ok, here's a question for the restore/ backup pros. 
On 1/4/14 I thought I would back up my iPhone 4s because it had been over a year (yeah, I know not a good idea to wait so long) since I had last backed it up (8/13/12, to be exact).  So, when I connected it to my computer, iTunes recommended I update the iTunes software.  So, I did.  Somewhere in the middle of the update I noticed that my iPhone kinda "rebooted" and I saw what a friend of mine referred to as the "restore iPhone" image on it.
Then, I received an error message from iTunes saying that there was a problem with my iPhone & that I would have to restore it to factory settings.  WTH!!!!
I went ahead and befored the restore.  Oddly enough, I seemed to get all my data back.  Heck, I even got emails & contacts that I added minutes before I tried to back it up.  I thought I had dodged a bullet & then I started to notice Apps that were missing & other Apps that I had deleted long ago back on the iPhone.
Then, other anomolies surfaced, like photos missing, contacts missing, & worst of all the App that I used to store some really preceious pictures, photo vault, was also gone (and all my most precious pictures).  Those pics were not in my camera roll any longer as I had moved them to this App.
Okay, I studied all I could on the net & learned where the backup files are stored on my computer and huh, it looks likes it took a back up right before (during?) the iTunes update that caused all this mess.  The problem is, iTunes doesn't show this backup as one I can select when trying to restore it.
My Question is, "How can I restore my iPhone to the backup dated 1/4/14 that I can clearly see in the backup folder?"
Many thanks,  psmcpher

Answer 1

oops.  typo:  "I went ahead and befored the restore."  should be... I went ahead and performed the restore.