http://armyservicenumbers.blogspot.com/ has a list of regiments with dates of joining for some service numbers.
For the Cameronians they have: 2921 joined on 11th January 1889. 3179 joined on 25th March 1890
So 3134 suggests an enlistment date around Nov 1889 to Jan 1890.
Ians1900 has cracked it! If I understand the Cardwell Reforms (Army Enlistment Act 1870) correctly, infantrymen would enlist for 12 years (previously 21 years) - seven in the Regular Army and five in the Reserve.
So, Arthur would have been Private 3134 in the Cameronians from c1890-95, transferred to the reserve c1895, worked as a labourer (as per 1897 marriage details), recalled to the Colours (same rank, number and regiment) probably when the Boer War kicked off in late 1899.
Philip