How have you tried to update the iPad to iOS 6.1.3 ? The page that you posted from, automatic downloads, isn't for iOS updates, only for content from the store.
If you have iOS 5+ on your iPad then you should have a Settings > General > Software Update option for direct update on the iPad, otherwise you can only update to the current version via your computer's iTunes.
To update via your computer, connect the iPad to your computer's iTunes and copy any purchases off it to your computer via File > Transfer Purchases (File > Devices > Transfer Purchases on iTunes 11). You may also want to copy photos and any important documents off the iPad as well e.g. via the file sharing section at the bottom of the device's apps tab when connected to iTunes, via wifi, email, dropbox etc - they should be included in the backup, but it's best to have a copy of them outside of the backup just in case. You can then force a backup of the iPad by right-clicking the iPad 'Device' on the left-hand side of iTunes and selecting 'Backup'.
Then start the update by selecting the iPad on the left-hand side of iTunes (on iTunes 11 you can enable the left-hand sidebar via option-command-S on a Mac, control-S on a PC), and on the Summary tab on the right-hand side clicking the Check For Updates button