You can take advantage of the split year basis of assessment, so you will get a full year's tax credits in both countries.
So you will pay no tax in Ireland.
Your employer should operate tax as normal, and you will claim a refund.
If the amount of the refund was not large, the employer might adjust the tax credits to give you two months' credits and lower rate bands. In theory this is wrong, but in practice it saves a lot of admin and I doubt if the Revenue would have any problem with it.