Adding different classes to anchor in navigation menu

Yes, it is possible.

You can achieve this using wp_nav_menu_objects filter.

function my_wp_nav_menu_objects($objects, $args) {
    foreach($objects as $key => $item) {
        $objects[$key]->classes[] = 'my-class';
    }
    return $objects;
}
add_filter('wp_nav_menu_objects', 'my_wp_nav_menu_objects', 10, 2);

The only problem is that these classes will be added to li elements and not to links directly. But it’s default WordPress behavior and I don’t think you should change it.

If you really have to change it, it is still possible:

function my_walker_nav_menu_start_el($item_output, $item, $depth, $args) {
    // you can put your if statements in here (use item, depth and args in conditions)
    if ( $depth == 1 ) {
        $item_output = preg_replace('/<a /', '<a class="level-1-menu" ', $item_output, 1);
    } else if ( $depth == 2 )
        $item_output = preg_replace('/<a /', '<a class="level-2-menu" ', $item_output, 1);
    }
    // .. and so on
    return $item_output;
}
add_filter('walker_nav_menu_start_el', 'my_walker_nav_menu_start_el', 10, 4);