Request a page - as admin - via command line to build up cache
Load the page via the cron (maybe 5 mins after deleting the cache) using wget. You'll need to set this up on the server as opposed to through the Drupal UI. 

This won't hit the page as admin but I'm not sure what value that adds. You just need the caches to be built so hitting as an anonymous user will do this.

Also checkout hook_drush_command() for creating your own Drush functions (good tutorial here).