How to add navigation arrows to manually slide through posts?

Your code suggests that you want a list of 5 most recent posts. There is no navigation for a list, unless you use paging. It is different, when you display a single post page. Then you can navigate to ‘next’ or ‘previous’ post, with help of the_post_navigation() template function. See it in Code Reference.

I’ve modified slightly your code, by adding 'post_status' => 'publish' to $args array, so you don’t need to check the status. Cleaned up incorrect nesting of HTML tags, as well. New code:

<?php 
    $args = array( 'numberposts' => '5', 'post_status' => 'publish' );
    $recent_posts = wp_get_recent_posts($args);
    echo '<ul>';
    foreach( $recent_posts as $recent ) {
        echo '<li class="one_sixth" style="list-style-position: outside;"><a href="' . get_permalink( $recent["ID"] ) . '" title="See ' . esc_attr( $recent["post_title"] ) . '" >' . $recent["post_title"] . '</a><br>Author: ' . get_the_author_meta( 'display_name', $recent["post_author"] ) . '</li> ';
    }
    wp_reset_query();
    echo '</ul>';