The "Rotunda, Rutland square" is a group of connected buildings on Rutland (now Parnell) Square, just beyond the north end of Upper Sackville (now O'Connell) Street where it leads into Cavendish Row. The complex housed the oldest maternity hospital in Ireland, and possibly the world, in addition to theaters for arts and entertainment, and rooms for hosting social and fundraising events.
The hospital building opened in 1757 as the new home of The Dublin Lying-In Hospital, which Bartholomew Mosse had founded a decade earlier in George's Lane. The architect, Richard Cassels, designed the large circular Rotunda, finished several years later, as an entertainment space, because the hospital was not-for-profit and needed to raise funds. Today, the Gate Theatre and the Ambassador Theatre reside in the complex.
Bloom passes the hospital on his way to the Glasnevin cemetery in Hades. Before his funeral train can get through the intersection another one passes in front, possibly bearing a baby who has died in childbirth at the hospital: "White horses with white frontlet plumes came round the Rotunda corner, galloping. A tiny coffin flashed by." In Sirens Blazes Boylan passes the Rotunda on his way to the Blooms' house, adding to the list of urban landmarks that both men see as they travel up Sackville Street: Sir John Gray's statue, Elvery's, Nelson's pillar, Father Mathew's statue.
Ithaca also mentions that Bloom has gone to see the circus in the Rotunda: "once at a performance of Albert Hengler's circus in the Rotunda, Rutland square, Dublin, an intuitive particoloured clown in quest of paternity had penetrated from the ring to a place in the auditorium where Bloom, solitary, was seated and had publicly declared to an exhilarated audience that he (Bloom) was his (the clown's) papa." The so-called "Round Room" inside the Rotunda proper was an ornate performance space, with beautiful Corinthian pilasters and stuccoed recesses. Because it had no interior columns supporting the roof, it could accommodate very large audiences, as seen in the engraving at right. In addition to popular entertainments like the circus, many important musical concerts were held there. See https://basilwalsh.wordpress.com/tag/dublin-rotunda.