Dromore

James Hannah came to New Zealand as a soldier of the 65th Regiment. The ship "Java"arrived in 1846. Hid records state he was born in Dromore, enlisted 1.1.1839. However, I do not know the county as both County Down Nad County Tyrone have a parish named Dromore. I've searched the internet trying to find when and where the 65th (2nd Yorkshire North Riding) regiment was in Ireland but haven't found any names of towns, or dates.