Julie:
I normally search on the subscription site Roots Ireland but when Co. Wexford may be involved I use the subscription site FindMyPast.ie Roots only has about half the Wexford RC parishes including in their data base while FindMyPast.ie has all of the parishes.
The unknown with John Bulger (besides not knowing his parents names) is that we can't be sure that 1839 was his actual year of birth since many immigrants changed their age when they arrived in their new country. In any event, I located seven 1839 baptismal records which are possible leads
John baptized August 29 1839 RC parish of Bree in Wexford father James and mother Sally Doyle
John baptized December 5 1839 (born December 3) RC parish of Wexford town father Morgan and mother Alice Murphy
John baptized September 30 1839 (born September 29) RC parish of Arklow which includes Wicklow and Wexford father John mother Margaret Brien
John baptized June 4 1839 RC parish of Mullinahone Co. Tipperary father Patrick mother Alicia Meaney
John baptized August 8 1839 RC parish of Arles Co. Laois father Daniel mother Margaret Bruder
John baptized October 25 1839 RC parish of Ballyadams Co.Laois father Michael mother Catherine Kelly
John baptized August 25 1839 RC parish of Tullagher Co. Kilkenny father John mother Ellen Kavanagh
I searched for Bridget Gallagher baptismal records from 1845-1855 in Co. Kilkenny and no records came up so not sure where to go next.
Roger