The update to LaunchPad 4.0.0.421 is automatic.

Character Select now works again in the EverQuest II LaunchPad!  Characters that have been logged into game within the last few months will be present in the "Select a Character" dropdown and displayed in the order of last played.  If the account has no characters on it, or you have no account logged in at all, nothing will be displayed.  If you have more than 5 characters in the list you can cycle through them via the NEXT and BACK options at the bottom of the display.

You can find the "Select a Character" dropdown in the upper right of the EQII LaunchPad.

Additionally there were a few bug fixes, the most obvious of which is that the "Remember Me" checkbox option is no longer persistant if you change between accounts.

These changes are automatically applied and you should see them the next time you open the LaunchPad.

source: LaunchPad Forums