A week in Dublin

We had 3 weeks in Ireland and Northern Ireland this fall, and our Dublin time was just 3 days. As you mentioned, the countryside/coastline are bigger draws. We stayed in a central Dublin apartment, and walked or took taxis, saw a play, visited some Trad Music pubs (avoid the Temple Bar scene, especially after 10 pm, when guys urinate on the sidewalks instead of using a proper WC), saw the Kilmainham Gaol, strolled the city, and went to a couple of museums. Three days was plenty for us.

We saved Dublin for the end of our trip, and used a rental car to visit sights throughout the northern stretches of Ireland and Northern Ireland, including Galway, Clifden, Trim, and the Neolithic Boyne Valley sights.

You can cover the highlights of Dublin in 2 or 3 days, but it's a good base for day trips as well. There are numerous tour companies that can get you to Powerscourt, Glendalough and the Wicklow Mountains to the south, and to Newgrange, the Boyne Valley and Tara to the north. It's about an hour and a half by train to Kilkenny, 2 hours to Belfast, and about 2.5 hrs. over to Galway, so you should have plenty of options to fill up a week in the city if you don't want to rent a car and strike out a little further afield on your own.
Check https://www.irishtrails.ie/Home/ for ideas on day hikes. Can easily take the DART to some of the nearby suburbs (Howth for example) to access the trailheads, then return to the city the same way once you've had enough walking.

I don't know if you could fill a week there, but I could. I went to the Archaeology Museum twice. Visit Dublinia for a hands-on experience and learn about the real vikings. Visit both cathedrals. Take a walking tour or two. There are two theatres, the Gate and the Abbey, with London-quality productions at bargain prices. For a fun hour visit the Leprechaun Museum, the guide I had (only by guided tour) was charming and funny and I learned a lot about Irish culture. The main post office has a small free museum worth seeing. If you are interested in history, visit the Collins Barracks. One of the best things I did was the Dublin Literary Pub Crawl. It's fun to explore the pubs, go in and wander from room to room. If you like the atmosphere, order a drink and chat with the other patrons. Some pubs have live music. In Temple Bar, they all have live music, some in the afternoons, most/all in the evenings. There are charming parks, and don't miss the Dublin doors.

Much of Dublin is easy walking, for farther points, there are buses and the Luas (tram). You absolutely do not want a car in Dublin, driving on the left, one-way streets, horrible traffic jams.

You can also take the train to Belfast for a day trip.