I think you mean Alfred Place/Street, which is now FITZROY CHURCH 2008
Fitzroy Avenue Church was built in 1872-1874 as the new premises for the congregation then located at Alfred Street (opened in 1837) and previously located at premises in Alfred Place (opened in 1821)
So who are his/her parents? Who are the witnesses? That sometimes helps. It looks like the parish records go back to 1821. Are you able to look through the baptisms, deaths, etc. to find relations? I now they are available at PRONI but still looking for somewhere on line that might have them.
You are very fortunate, presbyterian ministers were not allowed to solomize marriages until 1845 so many records of burials, marriages, etc were done by the Church of Ireland up until that time.