The Dep’t of Foreign Affairs has a Legalization Office that deals with the issue of authenticating documents.
Documents must be mailed, although you should contact them first through their online query form.
Before the COVID-pandemic took hold, you could visit the Cork or Dublin offices of the department. Unfortunately at the time of writing, only mailed applications are accepted.
Every document stamp costs €40, which can be paid by credit card or postal order. Expect the process to take about one week. If your teaching salary package includes visa application fee reimbursement, make sure to keep all your receipts.
If you happen to be living abroad and need to have your documents authenticated in Ireland, you can also use the services of a legalization agency. These agencies can act on your behalf and gather all the stamps and signatures that you need. Yes, it will cost you a little more, but it’s a convenient and it will save you time. Apostille.ie is one of several, well-rated agencies.
When authenticating your documents in Ireland, it’s important to always clearly explain what they will be used for. For the purposes of teaching, it’s for the China Working Visa. The department know about what the Chinese Government requires so they’ll be sure to use the correct wording/stamps/signatures. You will probably be asked why you need the docs authenticated but, if not, don’t forget to mention it.