It seems that the "Run as administrator" command only gives you some administrative privileges, and they're not enough to install Java (and OpenOffice, by association). In order to get around this, you need to actually log into the built-in administrator. Open a Command Prompt (Admin) with admin rights by using the Windows Key + X and selecting that option. At the prompt, type:
net user administrator /active:yes
Then log out of your regular account and back in as the built-in administrator, where you might be able to install the update. When you're done, log back into your regular account and, for security purposes, disable the built-in administrator account using: