How to add new post using a form to categories when categories are using as menu

by

The post_category parameter has to be an array, try this:

$post = array('post_type'=>'post',
 'post_author'=>$author,
 'post_status'=>'publish',
 'post_title'  => 'Test Title',
 'post_category' => array('679')
);

Ref: http://codex.wordpress.org/Function_Reference/wp_insert_post#Parameters

If that doesn’t work for you, try using category_name:

'category_name' => 'category_name',