Saving category to a post, before publishing the post

You can use the function wp_set_post_terms to set a category before the post is published. You need to get the post_id by call the global variable $post and get the id by $post->ID.

Here is a simple example. Change the id (2) to the id of your wanted category.

function wpse_78701_add_category_before_post() {
    global $post;
    if( $post->ID ) {
        wp_set_post_terms( $post->ID, 2, 'category' );
    }
}
add_action('admin_head', 'wpse_78701_add_category_before_post');

Update

If you want to change the category that will be saved when the user clicks on the link you have to add something like ?cat=2 on the dashboard-links like this:

echo '<a href="https://wordpress.stackexchange.com/questions/78701/post-new.php?cat=1">'. __('Add new post in category X', 'theme') .'</a>';

Then you can get the category bu use $_GET['cat']; like this:

function wpse_78701_add_category_before_post() {
    global $post;

    // Get category-ID from the link in dashboard (cat=X)
    $category = ( isset( $_GET['cat'] ) ? $_GET['cat'] : '' );

    if( isset( $post ) && $post->ID ) {
        wp_set_post_terms( $post->ID, $category, 'category' );
    }
}
add_action('admin_head', 'wpse_78701_add_category_before_post');