The Belfast Grand Opera House is a survivor. It opened in 1895 and has weathered decline in musical theater audiences, the threat of demolition, and blast damage during the Troubles (it was unfortunately near the often-targeted Europa Bar). It spent some time as a cinema and nearly went under in the 1970s, but now it's been renovated and is back in business.
It stands over the road from the wonderfully ornate fantasy that is the Crown Liquor Saloon, and it shares something of its aesthetic. Think late-Victorian splendor with lashings of gilt and fancywork. You can see more than opera here; it's also a popular venue for musical theater.
The Belfast Grand Opera House is located in the center of the city. Take a train to Great Victoria Street station.