how can i download an old version of youtube

If you have iOS 5 as your tagline says then you should have the built-in YouTube app, that was only removed in iOS 6

For the version of their app in the store, whether you can download it depends upon if Google allows redownloading of older versions of their app, and if they do if they have an iOS 5 version (which they possibly don't as until iOS 6 the built-in app was theirs). If they do, and the app is in your account's purchase history (if not then download the current version on your computer's iTunes), then see if you can download a compatible version via the Purchased tab in the App Store app. If not then you can access YouTube via Safari, or see if any of the other YouTube apps in your country's app store