How can I add category variable inside posts?

You can use get_the_category_list(), which returns a list of categories assigned to the post (each category in the list links to the category archive):

printf(
    '<span class="posted-on">last updated on %1$s</span><span class="byline"> <i class="fa fa-user"></i> %2$s</span> in %3$s',
    sprintf(
        '<a href="https://wordpress.stackexchange.com/questions/338246/%1$s" rel="bookmark">%2$s</a>',
        esc_url( get_permalink() ),
        $time_string
    ),
    sprintf(
        '<span class="author vcard"><a class="url fn n" href="https://wordpress.stackexchange.com/questions/338246/%1$s">%2$s</a></span>',
        esc_url( get_author_posts_url( get_the_author_meta( 'ID' ) ) ),
        esc_html( get_the_author() )
    ),
    get_the_category_list( ', ' )
);