How to get Post Type from the functions.php file

You have to create the function and call it when post ID is already set. Otherwise it will be empty. E.g. wp_head meets this requirement, init is too early.

function my_selective_wpautop() {
    global $post;

    if( is_singular() ) {
        if( 'page' == $post->post_type ) {
            remove_filter( 'the_content', 'wpautop' );
        } elseif ( 'post' == $post->post_type ) {
            add_filter( 'the_content', 'wpautop' );
        }
    }
}

add_action( 'wp_head', 'my_selective_wpautop' );