How do I get Link for Oldest Custom Post Type (dynamically)

Question

Simply change order parameter from DESC to ASC:

function Get_First_permalink(){
    global $post;
    $tmp_post = $post;
    $args = array(
        'numberposts'     => 1,
        'offset'          => 0,
        'orderby'         => 'post_date',
        'order'           => 'ASC',
        'post_type'       => 'POST_TYPE_NAME',
        'post_status'     => 'publish' );
    $myposts = get_posts( $args );
    setup_postdata($myposts[0]);
    $permalink = get_permalink($post->ID);
    $post = $tmp_post;
    return $permalink;
}

so once you paste that function in your theme’s functions.php and you have changed POST_TYPE_NAME to your post type name ,you can just call it whenever you want:

<a href="https://wordpress.stackexchange.com/questions/18647/<?php echo Get_First_permalink(); ?>">First Post</a>