It's so good to hear you're planning on celebrating both New Year's Eve and a birthday here on the island!
There are certainly plenty of wonderful pubs to stop by while in Dublin. There's an article right on the Ireland.com website that lists a few favourites
, and in addition, there are always truly famous spots like the Temple Bar.
For hours of operation, though, I'd recommend reaching out to these pubs directly, just to be sure of their holiday hours. If they do bookings, it might be a good idea to make reservations, as a number of people may be celebrating and the city could be busier than usual as a result. In addition, you may also want to take a peek at NYF Dublin
for more New Years' options that you may love.
I do hope this helps!