Query custom post type and custom field by URL parameters

To achieve this, you can start by implementing the solution for this question. That will allow you to filter posts by their first letter (no custom meta required):

Assuming starts_with is the name of the argument we want to use, we
can filter posts_where to add a WHERE clause limiting results to
those that begin with the given value if starts_with has been set on
the query:

function wpse_298888_posts_where( $where, $query ) {
  global $wpdb;

  $starts_with = $query->get( 'starts_with' );

  if ( $starts_with ) {
      $where .= " AND $wpdb->posts.post_title LIKE '$starts_with%'";
  }

  return $where;
}
add_filter( 'posts_where', 'wpse_298888_posts_where', 10, 2 );

With this filter added, we can query posts like this:

$query = new WP_Query( array(
  'starts_with' => 'M',
) );

That will return all posts beginning with “M”.

To support filtering the post type archive this way via the URL, you just need to register starts_with as a query var:

function wpse_346729_query_vars( $query_vars ) {
    $query_vars[] = 'starts_with';

    return $query_vars;
}
add_filter( 'query_vars', 'wpse_346729_query_vars' );

Doing this means that if the starts_with parameter is passed via the URL, it will be set on the main query, which will cause our posts_where filter to be applied to the current main query.

http://example.com/?post_type=staff&starts_with=A