I want to first off say that while on paper US Zip codes are more precise than UK Post Codes, the reality is often very different. The majority of people write their zip code without the "+4" which only gives you a city block, or roughly the same precision as a UK Post Code.
Secondly, I feel like Ireland missed an opportunity here. If they are only adding zip codes now then, yes, going with a US-style system (with the +4) is the correct decision. However why not invent a "person delivery code" rather than a "location delivery code" system?
I know, I know, crazy. But imagine this: Each address is assigned a unique code (e.g. starting with 0-9) and each person is also assigned a unique code (e.g. starting with A-Z) which is this mapped to a address when a letter is routed.
When someone moves house they give the post man a card, which says "Bob Smith (A123456) has moved from 87654 to 838449, please forward my mail indefinitely to that address."
That way when you move house your mail automatically follows you.
PS - I am suggesting that the "people codes" be an add on, so traditional "send to address" mail methods would work (e.g. sending "remember to vote" to every house on a street, rather than every individual would still work).
PPS - Calculating shipping would be no harder than today, just select your city and enter your "person code" or zip-code/house number.