Dublin Accommodations

The best castle near Dublin is the Rock of Cashel: https://heritageireland.ie/places-to-visit/the-rock-of-cashel/. Cashel is about two hours from Dublin.

Though there's no castle at Glendalough, a monastic settlement, it's packed with great antiquities: https://heritageireland.ie/places-to-visit/glendalough-monastic-site-and-visitor-centre/. Glendalough is only 30 minutes or so south of Dublin in the Wicklow mountains, which are beautiful. You can arrange a day trip to Glendalough at the Dublin Tourist Information office.

Newgrange and Trim are interesting, too.

Newgrange is neither a pretty village nor a castle. It is a Stone Age monument. Many refer to it as a passage tomb, but I'm not 100% sure that's its confirmed use. There is a very nice visitor center there and guided tours. Well worth your time. You can take an organized tour (Mary Gibbons, I want to say) from Dublin, and it is reachable by public transport.
Malahide is a popular day trip from Dublin and is home to a 12th century castle that you can tour. I don't know how pretty the village will be in March, but it's on the seafront. I had a nice pub lunch there a few years ago. Malahide is on the commuter rail line from Connolly. I have not been to Trim nor to Slane, which also has a castle.
I don't have a hotel recommendation, but I hope you're talking about next year because you might have a hard time finding anything, let alone anything affordable, two months out. I haven't been in Dublin on St Patrick's Day, but one tip I will give you is to avoid Temple Bar at all costs unless you want to be part of the massive, drunken crowd.