A push notification by definition is pushed from off system, but you can schedule local notifications without a network.
Since the countdown has a predictable time you can calculate when that is and schedule tile notifications to fire appropriately.
I'd skip toasts for this unless the user authorized them. If the countdown is an app feature the user wants then toasts are probably good. If the countdown is to the end of a trial or such then something less in-your-face is more polite. Either way, you should update the Tile even if you show the Toast since Toasts can easily be missed.
The app schedules tiles and toasts essentially the same way: create a TileUpdateManager or ToastNotificationManager , create a ScheduledTileNotification or ScheduledToastNotificaiton with the tile or toast information to show and the due date, then call the manager's AddToSchedule with the scheduled notification. When the due date arrives the tile and toast will be delivered even if the app is closed.
See How to schedule a tile notification and How to schedule a toast notification for walkthroughs.