iPhone won’t update nor install new apps
I seem to recall having a similar situation a couple years ago. I don't recall the exact steps I took to correct, but I would start by logging out of the App Store on the device and then logging back in. 

If that does not work, log out again, and then reset (not just restart) aspects of the device, then log back into the App Store. You didn't say what version of iOS you're running so you may have to poke around the settings a bit as Apple has changed how you do resets and what is affected. Make sure you have a back up of the phone first. With some of the new iOS versions, you can reset network settings before resetting all settings. I don't recall having to erase all data, so I believe you can skip that one. Again, please make sure you have a current backup!

With the iPhone5, if I recall correctly, you can press and hold both the power and home screen buttons for a few seconds which will also cause a reset.

After you try each step, I suggest just trying to download one or two updates at a time to make sure that will work. And I suggest being on a wireless network you have tested.

HTH,