How do I learn who wrote a (anonymous) 19th century Punch article?
As far as we know, a comprehensive author-title index to Punch does not exist prior to 1902 when the magazine started its own annual index. The Punch Archives in the British Library contain contributors' ledgers which identify authors. Please see http://www.ljmu.ac.uk/lea/114802.htm for information about a digital project underway to develop an interactive database of the contributors to Punch. You can also contact the British Library manuscripts' reference team here.
In the meantime, you can try searching the MLA Bibliography to determine if authorship of the item in question has been researched. The journal Notes & Queries is particularly good for that sort of question and is indexed in MLA.