Hello
I would contact both the City Cemetery and Milltown first, for a quick check. They can check for a burial easily, as long as they have the date of death. I found my gggrandfathers burial that way. They are very nice and helpful, and will call you back if necessary. Or post or email you also.
You can find their addresses and telephone or email by doing a search on Google.
If your ancestor is not buried in either of these, then you could check the local Carickfergus cemeteries. Do you know if Brougham St was in Belfast?
Good luck
Lucy