Karen, Hi!
There used to be a very large Convent in Sundays Well, which is itself quite a small area, (and I would imagine that this is the Convent the streetname refers to) named "The Good Shepherd Convent". I say "used to be" because it ceased to be a convent some years ago and then was burnt to the ground shortly afterwards, the spot now being occupied, I think, by apartments, although I am going on memory. I knew the convent well- it was very large indeed, had been around many, many years and was run by the Good Shepherd Nuns. Its history goes well back into the 1800s. It was basically an institution and housed a large number of women who had either mental problems or were what was then regarded as being 'fallen' ie pregnant outside of marriage. The women would work within the convent, and I am almost 100% sure that a Magdalen Laundry was run there. I was in there in the early 1980s on a visit, but my memories of the place are vague. I do remember it being a very busy place, with a lot of work going on. One notable and famous inhabitant in the early 1900s was a little girl known as 'Little Nellie of Holy God', a child noted for her holiness. She died at a very young age and her grave is in the grounds and is beloved of older Corkonians. For many years you could visit her room in the convent.
I am not in Cork at the mo, but was there last week and will be again in a month or so from now. There are Convent Places in other areas of Cork, which is something to bear in mind, two of them not a million miles from Sundays Well. In fact, the more accurate name for what is colloquially known as Convent Place in Sundays Well, is actually probably Convent Avenue. When I am next in Cork, I can try to search out the house for you and establish the streetname with more accuracy. I spend a lot of time around Sundays Well anyway.
Ceallachain