A simple question, but regretably a slightly complicated answer.
1881 was a momentous year for the British Army as it was the start of the Childers Reforms. The old numbered Regular regiments* were given territorial links and names, and the previously separate Militia battalions and the Rifle volunteers were all swept up in the amalgamation. The result was the SWB had two 3rd Battalions.
The first of these was the Militia battalion (mentioned by Shaun) formed from the old Royal Radnor and Brecknock Rifles. They continued to have that association with Radnor and Brecknock. The militia had very specific terms of service which meant they could only be called out ('embodied') by a proclamation issued by the Lord Lieutenant of the county and even then they could only remain embodied for a limited time. Their role was for local defence and once they had completed their initial training they only had to parade once a year and did very little continuation training. Men for the militia were selected from list drawn up by their local communities and they had to fulfill their 3 year commitment unless they were either grossly unfit for military service or they could pay a substitute to take their place.
The other 3rd Battalion was the Volunteer battalion formed from the 2nd Monmouthshire Rifle Volunteer Corps. These men were true volunteers and were consequently more efficient soldiers and were paid more than the militia. They were based in Pontypool and Brecon. A large number of the men of A, B and C Companies came from the workforce of the steelworks and tinplate works just outside Pontypool. By 1886 they were renumbered as the 2nd Volunteer Battalion SWB (not to be confused with the Regular 2nd Battalion SWB.) 28 Officers and 709 NCOs and men of the battalion volunteered to go the South Africa in January 1900 to fight in the Second Boer War. Three of the officers and 34 NCOs and men died in South Africa, either killed by the Boers, or succumbed to disease.
So depending on where the person you are interested in was living, you can probably work out which battalion he served with. The muster rolls for the 3rd Volunteer battalion have survived, but I'm not sure if the militia rolls have also survived.
If you search the (free) newspaper archive at the National Library of Wales (as suggested by hanes teulu), you will find many a reports about the activities of the 3rd Volunteer Battalion, their annual camps, social events and shooting competitions etc.
* the regular South Wales Borders were previously the 24th Regiment of Foot. A brief regimental history here:
https://web.archive.org/web/20060116064146/http://www.24thfoot.org/history.htm