If it's a Steam limitation to not show how many existing hours you've played (unless you've been using the steam launcher for it). How does unlocking achievements work if we're already eligible for them as you're saying they'll unlock?
The Steam Hours Played is a system that tracks hours played within the app with Steam open (ie. it's not a link in feature).
On the Achievement front, you can check elements within the game to unlock the Achievements (which in some of these cases is the same hook checked for the in game Achievement).