I am not an expert in this area but as I understand it this is the position.
You would have to file a tax return to Revenue Ireland showing; income, allowable deductions, taxable profit. This would then be taxed at the applicable rates based on your Irish income. VERY ROUGHLY about 25% up to about €36,000 of profit and about 50% on profit over that. The property is located in Ireland, your citizenship makes no difference.
If you are resident outside Ireland the tenant or agent is obliged to withhold part of the rent (20% I think) and pay it directly to Revenue.
After the first year you will also have to pay preliminary tax. Equal to the expected amount of tax due.
Neither withholding tax nor preliminary tax are a cost to you, the tax calculation determines the cost and then you get back any over payment. However they are a cash-flow issue. You effectively have to pay approx 120% of the tax in advance and then claim it back.
You will get it back once you have filed your tax return for the year, Revenue are generally quiet prompt about this.
Of course as a US citizen you will also have to pay tax to the US government on the rental income. This is in addition to the tax paid in Ireland.
There is a double taxation agreement which allows you to claim the tax paid in Ireland against your US tax, but as I understand it this is not perfectly efficient. In other words you might be partly taxed twice.
Proper professional advice on all this would be nice but unless you are dealing on a large scale it will cost.