Do Irish banks exchange USD for Euros?

Yes but they will probably charge you an arm and a leg i.e. a transaction fee and unfavorable exchange rate.
You might get your euros from your own bank if you have one before you go
or at a currency exchange in your town; so that way you are not trying to figure out which bank to go to in Ireland that offers that service.

Either way getting cash from an ATM abroad is cheaper.

thank you. I appreciate that information

Do not except in the most dire circumstances use a credit card in an ATM anywhere. It constitutes a cash loan rather than an account withdrawal and the interest rate resembles being slugged by a baseball bat.
Maybe you could meet your avoidance of ATM/debit cards half way. Don't use them for purchases, just to withdraw cash. It really is the cheapest way to change money. The fees involved are from your own bank (almost always) and can vary from institution to institution. I keep the charges minimal, on a percentage basis, by withdrawing a large amount and paying cash for all except really large purchases. Fast, simple, no easier to lose than plastic.
You can shop around for an ATM card with low, or if you are lucky no, foreign exchange fees. General opinion seems to be that pre-paid cards are more expensive. I always travel alone so I carry two ATM cards (and two credit cards) drawn on accounts at different banks. The cards are never all in the same wallet at once, for security and in case an ATM machine has the hiccups.

" I don't have an ATM card "

Do you have a checking or savings account? If you do, get an ATM card for it. Even if you just use it for European travel, this is BY FAR the best, easiest, and cheapest way to get cash while traveling. As someone said on another forum, there is no second best way; all the other alternatives have much higher costs, take much more time, and entail much more hassle.

Of course, learn about the fees your ATM card will charge you when you use it abroad. Many of the big banks, like Chase and Bank of America, charge $5 plus 3% for every foreign withdrawal. This is still a better deal than changing cash, but there are ways to beat these fees too.

If you don't have an account that can give you an ATM card, or if yours has high fees, you can open one online with Charles Schwab. I have no experience with them, but others have posted that these accounts have no fees, either for the account themselves or for using the ATM card abroad. More information here: http://www.schwab.com/public/schwab/banking_lending/checking_account

If you live in an area with TD Bank, they are also a great deal. Their Convenience Checking account, with a $100 minimum balance, charges $3 per foreign withdrawal with no other markup, and their Premier Checking account, with a $2500 minimum balance, charges nothing at all for foreign withdrawals. I have used them for quite a while and am very satisfied. Locations here: https://www.tdbank.com/net/absearch/

If you don't want to use an ATM, then your best bet is a local bank. Do not use a Change Bureau as they offer the worst rates.

Also check with your bank. Some U.S. banks have agreements with banks overseas to use their services for little or no fees.

A bank changing Dollars for you may vary based on their policies, some may only do exchanges if you are a customer with them or an affiliated bank, but worth trying. I would expect to pay in the range of 5% in fees or exchange rate for a $200 transaction from a Bank, closer to 10% from a Change Bureau.

As for Cash advance, understand your fee's and how to go about it. You will need a PIN from the credit card company, then you can use the credit card in an ATM for a Cash Advance. Verify with your card, but the most common cost for an ATM cash advance is $10 or 5%, which ever is more. Interest of ~24% starts from withdrawal, as opposed to a 30 day grace period on purchases. For the same $200 withdrawal given in the examples above, that would amount to: $200 + $10 + Interest. If you go online and pay off the cash advance immediately (Assuming you keep a zero balance) then you pay a $0.50 minimum charge, if you wait until your statement posts and use the money for a month, then interest would amount to $4, so your transaction cost you 5% to 7%. That really is not the "Sky is Falling......NEVER do a Cash Advance" advice that is commonly passed out. It is another option in your toolbox.

In fact an ATM withdrawal on your checking account is not always better, if you Bank at one of several big banks (like one of America or another famous for their stagecoach) for $200 you would be hit with a $5 fee and a 3% Foreign Transaction fee ($6 on $200) coming to 5.5%, the same as a cash advance at the low end, and about the same as exchanging cash here or there.

My final advice, if all you are headed there with is one credit card and some cash, I would urge you to open one of the accounts that were mentioned (Schwab, Capital One checking) or at least get a second credit card to have as a back-up. Just one source of payment is risky.

"A bank changing Dollars for you may vary based on their policies, some may only do exchanges if you are a customer with them or an affiliated bank, but worth trying."

I forgot to address this point in my post, but it's worth emphasizing. More and more banks will NOT do currency exchange if you do not have an account with that bank. So, you may be forced to go to a currency exchange bureau after all.

Is your plan to carry a thousand in cash (USD) to Ireland to exchange (no snarkiness intended)? Why not get an ATM card, this will be the least of your hassles compared to the effort of traveling overseas.

A couple of years ago I opened an online checking account with Capital One bank. I can move money into it from my regular bank account over the internet, but more importantly for this discussion it comes with an ATM card with no bank fees anywhere in the world. I normally keep a low amount in the account just to keep it open but before leaving on a trip I move my travel money into it.

Even though my regular bank is a US subsidiary of a Spanish bank (Santander) with branches all over Europe, I would be charged $5 every time I used its ATM card overseas, even at a European Santander branch. Having the Capital One checking account avoids those fees and the risks of carrying around extra cash.

With there being an ATM on every corner in every town in Europe these days, bank are getting out of the currency exchange business. The ones that still do offer it mostly as a service to their own customers. It has been a few years since I went to Ireland, but even then a few of the tour members could not find a single bank willing to exchange currency for them (maybe they picked the wrong banks?). There is just too much risk for them even with the excessive mark ups they charge for the service. There are more and more really good counterfeits they take the risk of losing money on if they get stuck with one. Even my own bank in the US where I have had accounts for decades will not do an on the spot exchange but rather requires you to wait 10 business days to get the amount credited to your account for any foreign currency you want to exchange with them.

Are you a AAA member? You might look into their reloadable Travel Money Visa Debit card. You bring cash to just about any AAA Office and they will issue it for you on the spot. Some fees involved when setting up the card. I'll be using this for my trip later this year to withdraw from ATMs as needed. Feel more comfortable with this rather than the ATM card tied into my actual bank accounts. Any leftover funds on the card (LOL) can be used at home just like a gift card. Have a great trip!

qwertymiss, I hope you'll report back after your trip on the usability of that card.