Poyntzpass is a very small village on the border between County Armagh and County Down in Northern Ireland. It is close tl Armagh City. Hence you will probably get a lot more information on the Northern Ireland forum. Remember you will need sterling pounds and not euros in Northern Ireland.
It is on the main cross border train line from Dublin to Belfast, and there is a local station there However, I cannot figure out if the station is actually still open. It is 120 kms north of Dublin, so is probably closer to Belfast.
Lots of previous posts on car rental. If you did not want a car, you could get the Enterprise train to Newry and get a bus from Newry to Poynty pass. However, this is run by 2 different 2 train/bus companies in 2 different countries with 2 different currencies, hence it is slightly complicated to purchase tickets. You can check out the Dublin - Newry train timetable and ticket options at www.irishrail.ie, and the Newry to Poyntz Pass bus timetable at www.translink.co.uk, Example: translink.co.uk/Timetables/…
The translink site is better.
You could also get a taxi from Newry to Pontzpass (15 kms), and you can also get a bus from Dublin to Newry. Check out www.buseireann.ie.
Iwould recommend a car in your case, even if you only hire it for 2 days. You could also get it from the airport, which is on the way to Belfast. Perhaps do it at the end of your journey, that way you can drop it off there.
I would suggest you split your post into 1 for information on how to get to Poyntz pass in the Northern Ireland/Armagh forum, and another on castles close to Dublin (note: there are plenty of castle types in and around Dublin, but some castles are really just big old Georgian/Victorian grand houses). You probably need to be more specific about what you are looking for. That way you will get what you are after.
Hope that helps a little...