How much electronics can I bring duty-free on a plane to Ireland from outside the EU?

As I understand it the rules depend a lot on the circumstances surrounding your travel to the EU (I think the rules for this are similar across the EU but i'm not 100% sure on that) and the purpose of the goods.

As JonathanReez says there is an allowance of E430 (E215 for under 15s). This is only for noncommercial imports though, if you intend to sell the goods after importing them then it doesn't apply. http://www.revenue.ie/en/customs/leaflets/pn1878a.html

If you are resident (or have recently been resident) in the EU and are returning home from a trip abroad then anything you purchased while outside the EU is subject to this relatively small allowance. Stuff you took out with you should be able to be brought back without paying any further duty/taxes http://www.revenue.ie/en/customs/businesses/importing/#section4 . For high-value goods it's wise to take evidence that it was bought in the EU (or previously imported legally).

If you are visiting the EU on a short visit then personal belongings intended for your personal use can normally be brought in on the assumption that you will take them with you again when you leave http://www.revenue.ie/en/customs/leaflets/cdpn46.html . Temporary imports are also possible for other types of goods but will often need to be explicitly declared. Items intended to remain in the EU will be subject to duty/vat if they are imported for commercial purposes or the total value is above your allowance.

If you were previously resident outside the EU for a significant period of time and are taking up residence in the EU then there are special rules that apply allowing you to import your personal belongings without paying duty/vat on them subject to some conditions. http://www.revenue.ie/en/customs/leaflets/pn1875.html

I'm not sure if your studies will count as "visiting" or "taking up residence" for these purposes. I suspect it will depend on the length of the course.

According to the Irish Tax and Customs website:

You are allowed to bring in goods (including gifts, souvenirs, perfume and clothing) free of duty, the combined value of which does not exceed:

€430 in the case of an individual aged 15 years or over

€215 in the case of an individual aged under 15 years

However note that in general, items intended for personal consumption (e.g. an expensive Mac laptop) are not a concern for customs, as it's presumed you will take the item with you when you leave the country. Therefore as long as your monitor is unboxed and you can reasonably explain you will need it for the duration of your course, no duty will normally be charged.

I would suggest you to bring the invoice with you, to show that this is not a brand new item. Then there is a good probability that it won't get even checked.