Need Help w/Step by Step for Restoring YouTube on 1st Gen iPad w/iOS 5.1.1
The Youtube app on iOS 5.1.1 will no longer work, even if you happen to install it. This month (April), Google deprecated an API which the legacy app used in favor of a newer one extending functionality to the HTML5 web/iOS/Android app versions which have since been released in the App Store.

Please refer to source for more information. This change was announced March 2014 - source

If you really want to try and make it work, as an actual answer to the question:


Download the iOS 5.1.1 IPSW file for the original iPad
Launch iTunes with your iPad connected
Back up your iPad
Option-Click (Mac) or Shift-Click (Win) the Restore button when viewing the iPad property page in iTunes
Locate and select the iOS 5.1.1 ipsw file downloaded in Step 1
Restore your device using the ipsw and, once completed, do not allow iTunes to update to iOS 6