I could not find a provider for An Post that would display shipping rates for your store. You could list your own rates for An Post for specific rate rages.
Here is an app that helps you notify customers of tracking codes and they do list An Post. https://apps.shopify.com/aftership You might want to check them out.
You can check out the Shopify Shipment Tracking & Notify app to handle An Post tracking. Once you have the An Post account credentials and API keys, the integration with An Post can be easily done in the app.
Once this is done, you will be able to automate your tracking process. The app automatically sends the tracking notification to your customers regarding the package. Also, you get an integrated tracking lookup page, by which your customers can track their product from your store.