Facebook for Windows Phone have got the much requested feature, push notifications. The app will have a new pivot titles push notifications under settings page.
You have three options to choose from to receive,
1) No Push Notifications
2)Toasts and Tile count
3) Tile Count Only
Also you can choose the items for which you can receive notifications,
Message, Wall Post, Feed comment, Feed comment reply, Photo tag, Photo comment, Tagged photo comment, Photo comment reply, Friend Request, Friend confirmation.
By this time, your marketplace will notify about the update, if not find the app in the marketplace, you will find the update button.