Custom post type in theme not appearing

The problem is most likely this part:

'capability_type'     => array('newsletter','newsletters'),

Try to remove it or adjust it accordingly.

You should also prefix your functions to avoid possible name collisions. So replace:

function post_type() {

to e.g.

function arcath_post_type() {

and similarly:

add_action('init', 'arcath_post_types');