Drush v3 had a --user flag (--user=1), however, this seems to be removed since v4, according to the website. How can I run Drush 5 as logged in as user 1?
drush topic core-global-options; you will see that the --user flag is still supported.
Of course, you do not really 'log in' to your Drupal site with Drush. The usual convention is that Drush commands should call API-level functions, which are expected to operate independently of the current user. (Drupal does access checks before calling the API functions.) There are only a few instances, mostly in contrib, where setting the user is necessary.