The "3rd Tipperary Brigade" is the name of an old IRA unit, so the timeframe would be the War of Independence, early 1919 to mid 1921. The unit was involved in the first attack of the War of Independence at Soloheadbeg , Co. Tipperary.
Many men captured after the rising were sent to prisons or internment camps in Britain, Patrick could also have been sent to prison during the War of Independence or the later Civil War.
The belt buckle and button are from the National Army/Irish Free State Army - so some time after the start of 1922.
Many records from the Easter Rising and War of Independence are available on the Bureau of Military History website run by the Military Archives - including witness statements, documentation relating to pension claims etc.
The Military Archives based in Cathal Brugha Barracks in Rathmines, Dublin hold personnel files for people that served in the Army - you usually need to prove your connection to the person being researched, e.g. with birth, marriage, census records etc