Magento 1.9 – How to See Error Page Previews

As stated in the comments... This is just my opinion. I don't think it's a good idea to have the error page (404 or 503, specially the 503) include website assets (categories, products or any other entities that might come from the DB). The page should be as light as possible.
Let's say there is an error connecting to the DB. You want the customer to see that there is an error, but if you try to show categories in it, you will have an error in the error page. (error-ception).
My suggestion is to use a static page, with resources (images) served from and other server. A simple image with an unplugged computer should do the trick.

But if you really insist, here is a quick and dirty way to do it. Create a file (menu.php) where you create an instance of Mage::app() and generate the menu and include that in the error page.

Internal Server Errors and/or Gateway Errors can almost always tracked back to one or another PHP error. Turn on your logs and see what you can find from there.