Would each post only be in one category?
You’d first need to get all the categories associated with the post, and then output only first. Here is an untested example:
function add_before_content($content) {
$post_cats = get_the_category();
return '<p>This post is in the ' . $post_cats[0]->cat_name . ' category</p>' . $content;
}
add_filter('the_content', add_before_content);
Reference: http://codex.wordpress.org/Function_Reference/get_the_category