How to add a CSS class to the tooltip on a navigation menu?

Add this code into your functions.php

class Tooltips_Walker extends Walker_Nav_Menu
{

    function start_el(&$output, $item, $depth, $args)
    {
        $classes     = empty ( $item->classes ) ? array () : (array) $item->classes;

        $class_names = join(
            ' '
        ,   apply_filters(
                'nav_menu_css_class'
            ,   array_filter( $classes ), $item
            )
        );

        ! empty ( $class_names )
            and $class_names=" class="". esc_attr( $class_names ) . '"';

        $attributes="";

        if(strstr($class_names, 'tooltips')) {
            $attributes=" class="tooltips"";
        }
        $class_names  = str_replace('tooltips', '', $class_names);

        $output .= "<li id='menu-item-$item->ID' $class_names>";



        ! empty( $item->attr_title )
            and $attributes .= ' title="'  . esc_attr( $item->attr_title ) .'"';
        ! empty( $item->target )
            and $attributes .= ' target="' . esc_attr( $item->target     ) .'"';
        ! empty( $item->xfn )
            and $attributes .= ' rel="'    . esc_attr( $item->xfn        ) .'"';
        ! empty( $item->url )
            and $attributes .= ' href="'   . esc_attr( $item->url        ) .'"';

        // insert description for top level elements only
        // you may change this
        $description = ( ! empty ( $item->description ) and 0 == $depth )
            ?    esc_attr( $item->description )   : '';

        $title = apply_filters( 'the_title', $item->title, $item->ID );

        $item_output = $args->before
            . "<a $attributes title="".$description."">"
            . $args->link_before
            . $title
            . '</a> '
            . $args->link_after 
            . $args->after;

        // Since $output is called by reference we don't need to return anything.
        $output .= apply_filters(
            'walker_nav_menu_start_el'
        ,   $item_output
        ,   $item
        ,   $depth
        ,   $args
        );
    }
}

And update this wp_nav_menu code where your menu is called from theme [ header.php ]

    wp_nav_menu(
                    array (
                        'menu'            => 'main-menu',
                        'container'       => FALSE,
                        'container_id'    => FALSE,
                        'menu_class'      => '',
                        'menu_id'         => FALSE, 
                        'walker'          => new Tooltips_Walker
                    )
                );

and use “tooltips” – as menu item class and put the description to show as tips title.

and make sure the “Zebra_Tooltip” Css class z-index is more then all element’s z-index: or add this css code in style.css

.Zebra_Tooltip{ z-index: 100000 !important;}