If you are in Dublin in the National Archives they have ledgers relating to the individual National Schools in the 19th/ early 20th century. These ledgers do not contain a lot of useful information for the average teacher – mostly just a name and date of start / cessation of employment. But they may help you determine if your ancestor taught at a particular school. If there was a particular problem with his employment then he may well be covered in more detail, and this information is contained in separate files.
I found detailed correspondence in the files there related to my own great grandfather and the details leading up to his threatened dismal from the headship of Glanworth National School in North Cork. We knew from family stories that he had a problem with the drink. From the files it was apparent that on occasion this caused slurring of words etc. in the classroom. His written response to accusations of drunkenness, were impressive, and reading the files, without the personal knowledge, one might have believed his tales of neuralgia and the effect of medication taken for his complaint.