Module to list menu_router / path debug info?
I'm not sure there's a module specifically for that, if anything I think it would come under the domain of Devel as you mentioned in the comments. I haven't seen anything like that in Devel before though.

Getting the router item for the current path itself is very easy:

$item = menu_get_item();
$callback = $item['page callback'];
// etc...


You could just wrap that in a hook_init() and dpm() it out for a quick solution, or maybe build a patch up for Devel.