Check get_post value after wp-admin login

You just use a normal if statement around the notice so that it only appears if the value of your option matches the current date.

function wpse_277689_admin_notice() {
    if ( get_option( 'expiry_date' ) === date( 'd/m/Y') ) :
        ?>

        <div class="notice notice-success is-dismissible">
            <p><?php _e( 'Done!', 'sample-text-domain' ); ?></p>
        </div>

        <?php
    endif;
}
add_action( 'admin_notices', 'wpse_277689_admin_notice' );