How to update the App Store app on my iPhone 5?

I got a new iPhone 5 yesterday as a warranty exchange, and the app store is the old version from several months ago.

Is there a way to get the app updated soon, or do I have to wait several days for the app to update on its own?

Well there's one thing you could try. Settings>General>Software update. You will probably get IOS 7 if there's an update. Hope this helped!

App Store gets updated when iOS gets updated. As long as you are running the latest version you should have the latest App Store.

It can only be updated with iOS updates, individual updates for app store are not available for iPhone