Where is your employer resident? If in the UK your employer will have to operate UK PAYE, however, if you carry out more than 60 workdays in the Republic, your employer will also have to operate Irish PAYE. This is a complex situation and your employer will need to get tax advice on the correct operation of PAYE.
However, If your employer in resident in Ireland, they will operate Irish PAYE on your income, but if you are tax resident in the UK you will have to declare this income in the UK and get a foreign tax credit for any Irish tax paid by completing a tax return each year.
You can be paid either in euro or sterling, the currency doesn't matter, but beware currency fluctuations.