Post date on Custom field

There is a hook called save_posts

so you need to hook a function into that post like

add_action('save_posts','store_date');
function store_date(){
  if(get_post_type()=='post'){
   $month=get_the_date('M');
   $year=get_the_date('Y');
   update_post_meta(get_the_ID(),'month-field',$month);
   update_post_meta(get_The_ID(),'year-field',$year);
 }
}

This should be all you need

Here are additonal links to save_posts and update_post_meta