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.

Are you interested in a defensive castle or a manor house? If a defensive castle, Trim Castle is probably the easiest to access from Dublin. It's been "renovated" to demonstrate how it's use evolved over the centuries. The town of Trim also has a monastic ruin and the Yellow Tower. The town sits along the banks of River Boyne. Malahide Castle is really a manor house with beautiful gardens located on the coast with a nice high street and some very good restaurants. Kilkenny Castle is also a manor house in a quaint and charming village. The Rock of Cashel is a castle ruin with historic significance and well worth a visit but may not be the castle you're looking for.

What's your price range for lodging in Dublin?

In terms of accommodations, I’d recommend beginning by just cross referencing reviews in Booking dot com and Trip Advisor (Trip Advisor alone is no longer a reliable source but I’ve been pretty happy with places I’ve found with consistent good reviews across sites). Use the sites for your research but then book directly with the actual hotel.
If you’re looking to be centrally located, proximity to Saint Stephen’s Green is something to consider. However, Dublin is a relatively small and very walkable city so you can also consider further away, even across the river.
Without knowing your price range and nonnegotiables, it’s hard to recommend a hotel. We stayed at the Drury Court Hotel last May and liked the location, room cleanliness, and friendly staff. We didn’t opt for the breakfast package so I can’t speak to that. There are definitely more upscale hotels nearby, as well as hotels with a lower price point (although lodging and food in Dublin are both surprisingly expensive), so if you share more information here about what you’re looking for people will be better able to help.

We stayed at NYX Dublin Christchurch and found it well situated. It's not far from the Temple Bar area but without the noise. We found it to be walking distance to a number of sites and for those a little further out (i.e. Kilmainham Gaol) it was a short and direct bus ride.

When I visited last June, we day tripped to both Trim Castle and to Brú na Bóinne (site of Newgrange, Knowth and Dowth) and did it ourselves with a rental car and had a great day out. We chose to do that as we couldn't find any group tours running that visited both sites. It was a lovely day out and we were glad for it. On our way back to Dublin we stopped by the very quaint fishing village of Skerries and enjoyed an incredible seafood dinner at Stoop Your Head as recommended by someone on TripAdvisor and it was a picturesque way to round out the day.

We stayed three nights at The Davenport and enjoyed it greatly.