With Minimum cache lifetime, do you need to run cron.php

I think you are confusing two issues. The minimum cache lifetime just specifies how long old content will "resist" a call to cache_clear_all() and still be displayed to anonymous users. The typical way that cache_clear_all() is invoked is when a user submits a new post or comment.

While cron might cause the cache to be cleared, it usually won't. If you changed a module like this, there is no way for the Drupal core to know that there is something different about the pages, and hence the old content will be held in the cache.

For a site where I have the page cache enabled, I have a very small utility module that uses hook_cron to call cache_clear_all() every time cron is run so that the items in the event module calender, event block, etc. get updated and aren't days out of date if no one has posted new content that triggers a cache_clear_all().

This should be a functional module (for Drupal 4.7), save as zmod.module without the ending ?>

/**
 * Implementation of hook_help().
 */
function zmod_help($section) {
  switch ($section) {
    case 'admin/modules#description':
      return t('Simple utility functions');
  }
}

/**
 * Implementation of hook_cron().
 */
function zmod_cron() {
  //Clear the cache when cron.php is executed, at most every 5 minutes 
    $last = variable_get('cron_last', time());
    if ($last < time() - 300) {
      cache_clear_all();
      //uncomment the line below if you want to see notices in your log
      //watchdog('cron', 'zmod cleared the cache', WATCHDOG_NOTICE); 
    }
}