Some people are experiencing a problem with the time and date not syncing correctly on their new iPhone 5. It seems that this problem is mainly on Verizon’s network according to Apple Informer via Apple Insider. When Verizon was posed with the issue they claim that the problem is with Apple’s firmware. Apple is then pointing the finger at Verizon saying that the firmware is fine and that the problem is with Verizon’s network. At any rate the consumer is the victim and no one wants to take initiative to fix the issue. When I received my iPhone 5 I had a similar issue with the time and date not synced even though it was set to automatically sync via the network. So here is what I did to fix the problem and hopefully it works for you too.
Go into the Settings app>General>Date&Time. Once here turn Set Automatically to OFF and then manually choose your Time Zone and set the date and time. When that is completed turn the Set Automatically back to ON. After doing this the time and date on my phone was corrected.
Now if this doesn’t work for you, there are a couple of options to try. 1st repeat the same steps above while connected to WiFi, 2nd you could try a hard reset by pressing and holding the Home and Sleep/wake buttons once the Apple logo appears you can let go. Last, there is always a restore try this only if any of the above does not work. Good luck and I hope this helps.