How much cash to withdraw 1st day in Ireland

Not sure if you are quoting that statement within context and correctly. I generally withdrawn what I anticipate needing for the next three or four days but always plan to keep a 100 euro in reserve. Hidden and separate from other money so the first withdrawal always includes that the pad. I try to hit the ATM about twice a week and always on Friday before the weekend. Try not to have to use an ATM over the weekend. I have but prefer not to. Maybe the first withdraw could be four hundred euro. Less if paying hotel with credit card.

There are a couple of factors to consider. What does your Bank charge you for a withdrawal? I have two cards that charge the minimum network fees of less than 1%, so I can choose my amount with no penalty, so I will generally take out 200 to 300 euro just for meals, transportation, and spending money; more if I am paying cash for a hotel. If your bank is one of those that charges you $5 plus 3%, then it makes sense to get as much as you think you need in the first transaction. Myself, since I incur no penalty, I take out smaller amounts just as a safety measure, less risk of loss or theft.

Be aware that some ATMS have a withdrawal limit also. I requested 300; it said I couldn't have 300, but did I want 200 instead? .....not in those exact words. :)

Paul makes a good point. Different cards can have different fees or different withdrawal limits. My main card is a debit card from my local credit union, which charges nothing at all. My backup card is a debit card for an online bank. It charges 1%. If your card charges something more, particularly if it charges a per-use fee, take out as much as it will let you each time. If not, just get enough for a couple of days at a time.

Your own bank sets a daily withdrawal limit (often around $300, which you can ask to change) and any single machine has a per-transaction withdrawal limit (but you can make more than one transaction). I would withdraw my daily limit, with the knowledge that I'll use it eventually. Toward the end of your trip is when you'd be more careful not to take more than you expect to spend before leaving.

I usually withdraw €480 or €530 on my first withdrawal, so that I am sure of getting at least one €10 and one €20 bill (if you pick an even amount like €500, there's a chance you could get only 50's or 100's, which can be hard to use for small amounts in some situations). However, my bank has a high maximum daily withdrawal ($730) - much higher than most others. I then put about €50-80 in my wallet, and stash the rest in my money belt (I use the "hidden pocket" style that hangs off my belt, so I don't have to disrobe to get to it).

So, first find out what your bank's daily limit is, and if it seems low, ask if you can have it raised. Then, convert it to euros (just Google, for example, "450 USD in EUR"). You might want to write this down - the machines in Ireland will only ask how many euros you want; they won't show conversions rates, and if you exceed your bank's limit, the transaction will be rejected, but it still won't tell you your bank's limit. Similarly, the machines won't tell you their own limit, so if your first transaction is rejected, just ask for half the amount the second time. Different machines will have different limits. I ran into this in Israel - the bank running the machine in the airport arrivals area only allowed 1000 shekels, but machines from other banks would allow up to 2000 shekels. Again, none of these limits were on the machines themselves - I only found out from our regular poster Chani, after the fact.

Harold makes an excellent point about choosing an odd amount. I did not think of this and ended up with a handful of 50s. Small establishments will not take them, can't make change, unless you're buying something close to that amount.
We also took two debit cards on different banks, just in case one gave us trouble. One was a better deal, so we used that one when possible, but tested them both on arrival with a smaller amount just to be sure they both worked.