Hi,,,there is a service record for a Patrick Byrne b 1855 Baltinglass,,,,he enlisted in Armagh in 1877,,,,served till 1882,,,he was a groom by trade,,,gave his next of kin as his father Patrick.
His record states he was signed off due to debility. He was in Aldershot in 1881,,according to his sickness record,,,so he may be the Patrick Byrne in the 1881 census in the army at Aldershot ,,,,search for Patrick Byrne b 1856,,,then choose the result in Hartley-Wintney/Hampshire.
if this is your man,,,maybe ,,as he was a groom,,he worked with horses in the army,,,then later volunteered to work with them again in a civilian capacity in Australia.
What is the source of your info that he was in Australia working with horses ?