Convert With Custom Post Type

I think all you’d need to do is change the WP_Query from

$catquery = new WP_Query( 'cat=" .$category_id. "&posts_per_page=100' );

to

$catquery = new WP_Query( 'post_type=property&posts_per_page=100' );

This assumes your new custom post type is registered as “property” (lowercase) – if you registered the new post type using ‘Property’ or ‘properties’ use that instead.

UPDATE – trying to debug why the above isn’t working.
Try running this code to see if the query is returning the correct results. Run just this code, none of the rest of what you originally posted. This should display an array of all the results (properties) that are returned by that query, so we can make sure that part’s working correctly.

$catquery = new WP_Query( 'post_type=property&posts_per_page=100' );
var_dump($catquery);