Widget Options Not saving

Your problem is that you’re calling the wrong method for generating the name attribute. It should be $this->get_field_name('via_cat')

Also, you can use wp_dropdown_categories(), no need to reinvent the wheel:

wp_dropdown_categories(array(
  'name'      => $this->get_field_name('via_cat'),
  'selected'  => (int)$instance['via_cat'],
));