While I'm happy to try to support users of Achievements, I can't get back to you any quicker or slower than I can anyone else. Lately I've been able to respond pretty quickly, but everyone should expect responses from me within ~2 weeks of posting here.
The answer to your question is... not easily, because it hasn't been built. You would need to write a bunch of custom code to send an AJAX request to WordPress; then a receiver for WordPress to listen for that request; and then get it to clear a setting in one of the database tables. If you are (or have) a developer and can handle the first parts, I'll let you know what you need to do with the database to clear the popup.
I suppose alternatives would be to stop the popup from appearing all together, or perhaps customise its behaviour and template with your own CSS & JS.
If you want to stop the popup from appearing altogether, put this in your theme's functions.php:
add_filter( 'dpa_user_has_notifications', '__return_false' );