Sure, it's possible.
For the CSS (i.e. background image), you can link to your static site's stylesheet just by adding a link to it in your WordPress header.php.
For the navigation menu, you can put it in a separate file and call it in via an include in both your static site, and your WordPress site.
Example - you could put it in a file called menu.inc.php (or menu.php). (I recommend putting includes in a folder called includes to keep your files organized.)
Then in your static site, use a PHP include where you want to display your menu with a line like:
<?php include($_SERVER['DOCUMENT_ROOT'] . "/includes/menu.inc.php") ?>
The first part means that the link will be root-relative, meaning that it will work on a file in any directory of your site.
Put the same line in your WordPress header.php file where you want the menu.
I notice you are not using .php extensions on your static site - you'll either need to change that, or add a line to your .htaccess file to cause HTML files to be rendered as PHP. Since your site isn't fully built yet, I recommend changing extensions from .html to .php.
Let me know if you need any of this clarified or have any questions!