Please check this solution which worked for me since the ios 6.1 update. You dont need to restore anything, just a few minutes of your time will do. Try to do this proceedures while the phone is plugged in, and charged to 100%.
1. Go to Settings>Mail, Contacts, Calendars>iCloud
2. Turn off everything from mail to find my iPhone, but do not delete anything. Leave everything in your phone or iPad.
3. Follow the same proceedure for every mail account you have.
4. Go to Fetch New Data, turn off Push, and check Manual. Go to Advanced at the bottom, and swtich everything to Manual.
5. Go to Privacy>Location and turn it off.
6. Set Brightness to Auto
7. Kill all open apps. Reset the phone by holding both Power and Home buttons at the same time until the Apple logo appears.
8. Go back to Settings and turn your settings back ON except for Push.
For my iPhone 5, I have my wifi on 24/7. I only turn on data when i need it as it is really gulps you battery juices. You won't fix it the first time, just try to do it when you feel your battery is not right or when you feel your phone or other iDevices getting hot. Hope this will help you.