Does it give you an error reason? Can you log in to iTunes on a computer? Did your iTunes account password change or get security hold? Seems more like iTunes issue, not 5S issue.
Applications are forever tied to the Apple ID used to purchase them. If you try to update, do you get a request for a password for another Apple ID?
if you don't have the password, you will have to delete and repurchase with your current Apple ID.
This is the problem I am having- I was gaifted a (used) phone and my snapchat and facebook apps are tied to her old email. I factory restored the phone when I got it and wiped it clean...everything else will update fine, just these two apps that she previously had. Do I need to delete them and re-install?