William Patrick Mulvany

1784 is very early for Catholic records, not all parishes have records going back that far especially parishes outside the city.

I would treat ages at this time as estimates, very often incorrect so I'd allow at least 5 years either way for any search.

If there were any additional details from later records that gave a more accurate place of birth e.g. town or parish then research could be focused on those parishes, e.g. do records exists for this parish and time-frame, are they legible/faded torn etc. Another possible path that can be worth exploring is tracing a younger sibling if known, as later records are more likely to exist.

Catholic records for most of the parishes in the city are available on the free IrishGenealogy website, exception being St. Pauls, Catholic Registers for the vast majority of parishes in Ireland are available to browse (image only) on the National Library of Ireland website and have been indexed by both FindMyPast (you can use a free account for these..) and Ancestry - for faded or hard to read registers it may be necessary to browse page by page - a big task without knowing which parish to focus on..

One detail to bear in mind when searching is that names in early Catholic registers are often recorded in Latin - e.g. Gulielmus for William. Middle names do not appear very often on baptisms.