There is a glitch and I've yet to see an easy way around it. If you normally sync to a computer, you can download and install your updates via iTunes on that computer.
I tried to purchase an app and was asked my security questions since iTUnes didn't recognize my device. (which it should have because it did before iOS7). I could only answer one right and then when I tried to reset my questions the mail never came so I ended up having to call Apple and have them manually reset my security questions.
So maybe try that, find a free app, try to install it, go through the verification (and make sure you have/know your secret questions) and see if that frees things up. After I reset my questions, I can now download app updates normally.
It's like Apple's servers 'forgot' a lot of devices and until you force the servers to remember you, there's this limbo between 'we know you need up dates' and 'who the heck are you? I don't know you' and people get caught in the middle.