Quick solo trip- Galway, then not sure

We just spent last month in Ireland based near Newgrange in a house. We used the train to go from Drogheda/Dublin to Galway for similar day trips that you’re doing. We used the long distance buses too but we had a car for our nearby, local trips. If you want to see Newgrange without a car I suggest you stay in Dublin and book a tour from there. There aren’t very many other towns nearby with reliable transportation to Newgrange. Also when we were there last month the tickets were still scares but I checked the official site and tickets remain for your dates. https://heritageireland.ie/places-to-visit/bru-na-boinne-visitor-centre-newgrange-knowth-and-dowth/

If you want somewhere to go after Galway that Citybus has service to, check a route to Cork or Adare before your night(s) in Dublin to see Newgrange.

Also check on tour availability to Newgrange from Dublin. I do know of a 2 Inns near Newgrange and the town of Slane is nearby but you’d need a taxi to and from Newgrange.

If you are going to Newgrange by public transport then the bus leaves from Drogheda Railway Station, so Drogheda is the obvious town to research to stay overnight.

The first morning bus leaves Drogheda MacBride Station (via the Bus Station) at 0900 with buses back at 1220, 1540 and 1705 for the 40 minute journey.

Transport for Ireland will give you full details.

If you come by train from Galway to Dublin you will need to use the LUAS tram to cross from Heuston to Connolly station for the northern suburban trains to Drogheda.

If you come by bus the easiest way is bus to Dublin Airport direct then bus 100X at 20 minutes past each hour for the 40 minute ride to Drogheda (the 100X is headed for Dundalk). You could then stay at a town on the coastal railway such as Malahide for instance- Malahide has a direct bus to Dublin Airport every 30 minutes.

If you take the train from Galway, you could stop in Athlone. I haven't been, but it's on my list. Castle, Sean's Pub (is it the oldest in Ireland? who knows?). As others have noted, the train comes in to Heuston Station from the West, so you would have to transfer. If you decide to stay in Drogheda, I recommend the Marcy Hotel. I had a bit of street noise when I was there, but it was summer, and I had the window open, which you'll not want to do this time of year. Maybe ask for a room at the back. It's a little bit of hoof from the bus station (much further from the Drogheda train station, I think). Loads of bars and restaurants around, including a bar downstairs. The bus from Drogheda to the airport is easy and cheap.

I would also suggest a stop in Athlone. The castle is interesting, and you may be able to find a way out to Clonmacnoise from there.