You need to search Michael's civil and church wherever he emigrated to find specifically where he was born in County Mayo and his parents names. There's a list of the names of all the towns and townlands in Mayo at http://www.seanruad.com
and pick Mayo as your search county.
You need the specific birth location because church parish records for his religious denomination and birth location in Mayo are the possible source for a baptismal record for him. Not all church parish records survive back to 1843 when he was born.
There are no surviving census records before 1901 and civil registrations, bmds, vital records don't start until 1864 for births which is too late for a civil birth record for him.
If Michael was Roman Catholic, here's a link to the Roman Catholic parish maps in County Mayo. Click on each parish on the map to see the dates covered in the records and where the records can be accessed, most are available on order through LDS:http://www.ireland.com/ancestor/browse/counties/rcmaps/mayor...
Where did Michael emigrate to? Did he marry in Ireland? Was his daughter Annie born in Ireland or elsewhere?