This narrows the problem to your computer, but doesn't identify why your computer is not able to back up. Assuming the backup restore succeeds, you will need to connect to your original computer to sync your apps back, or copy your iTunes library to the new computer before syncing with it.
Alternatively, you can sync your music and content using the iTunes and App Store apps directly on the computer it works with.
I did a Restore NEW and Restore from Backup (rather, a couple of times)...but nothing seems to resolve the issue...Apps keep crashing, my iPhone 4 heats up so quickly and causes inconvenience to my ears when I take calls....All these have been happening immediately after iOS 6.1.3 upgrade...nothing seems to work, Lawrence...I suspect there is not any chance of a hardware issue!!! What do I do now???
Restore as NEW. Do not add any apps. Do not add any email accounts. Do not launch any buit in apps. Do not log in to the iTunes store. Do not log in to Game Center. Do not enable Genius. Does the battery still drain rapidly? If so you have a hardware problem. If the battery life is normal then something in the apps or settings that you make is causing the problem.