Go to this link and it will walk you through all the steps https://www.usps.com/international/passports.htm
There are recommended hotels on the Travel tab where Michael and I blocked off some rooms for you all.
Shannon (SNN) is an hour drive to the Castle but Dublin (DUB) seems to be cheaper.
Airfare $900/ person (depends on the day of the week you fly in and out) Hotel $100/night Rental Car $40/ day (we recommend getting a WiFi hot spot instead of GPS) Food & Drinks $40/ day / person