Hi Caren
If nobody holds additional information on this family you will have to rely on church baptismal and marriage registers to confirm birth/marriage details of Michael Cain. If you have not already done so you should check church registers for Dublin city, free of charge, at
http://www.irishgenealogy.ie .
You should also consider an examination of the all-Ireland database of church registers transcriptions at
http://www.rootsireland.ie . For example an examination of the indexes to this database reveals the birth of a Michael Kane in County Dublin in 1760. Kane and Cain are variant spellings of the same name. This, of course, is not sufficient evidence to confirm relevance of this Michael Kane to your family history.
Although RootsIreland is the largest online source of Irish church register transcripts, it must be emphasised that a failure to find relevant birth/marriage entries in this database (or indeed any of ‘big’ Irish databases) doesn't mean that the events you are looking for didn't happen in Ireland. It simply means that they are not recorded in the database; for example, they may be recorded in a record source which doesn't survive for the time period of interest or in a source that has not been computerised.