Filter the_content to add something before each element?

by

Just filter 'the_content', test if you are on the correct page, then run str_ireplace():

add_filter( 'the_content', 'wpse_76808_filter_h1' );

function wpse_76808_filter_h1( $content )
{
    // page slug
    if ( ! is_page( 'privacy-policy' ) )
        return $content;

    return str_ireplace( '<h1', '<a href="#">Back to the top</a><h1', $content );
}