So, I'm using my wordpress with a static front page and then I have Blog link that opens the blog posts.
My question is, how can I apply a different template/styling only to that area?
Refer to the Template Hierarchy in Codex.
If you have assigned a static front page, the template that will take priority for that page is
front-page.php, and the template for the posts (blog) page is
You can create a custom template for every page and assign it to that page from edit page screen in the admin.
WP Codex has an example about custom page template for "page of posts". That should fit your needs.
As for just targeting different pages with CSS you can add a
body_class() function which prints out a class for body element depending which page is displayed. Add it like this:
<body <?php body_class($class); ?>>
More information about
body_class() function: https://codex.wordpress.org/Function_Reference/body_class