To approve all pending posts at once is possible using WordPress core functions in on the wp-admin/edit.php page:
1. Determine the approx number of post to act on
2. Open Screen options and set the number of posts per page to this number
3. Show Pending only by the header action link to this filter
4. Mark all by checking the box to the left of the Title header
5. In mass actions select Edit
This will open a multi-post edit box. All fields should remain as "Don't change" except Status. Set the Status to Published. Update.
(Set the number of posts per page n Screen options back to a reasonable number, like 20.)