Do you know where he lived, where his child died, etc? What is Michael's birth year, based on the child's death? It must have been a minor child, living at home, for the parents' ages to be included, correct?
Who did Michael marry? Is there any hints that they might have been from the same part of Ireland (assuming his wife was Irish).
If you can tell us more of what you already know and where and why, it might be possible to work backward and find him.
Do you have his death record? That and his marriage record are the most likely places to find his parents' names, which will be essential if you want to trace him back to the Irish records.
Failing that, did he have siblings and have you looked for their records?
And there is a very longshot that his children's death records might say something beside "Ireland" for his place of birth. Have you tried them?
I found 4 Michael Ryans arriving in 1892, but none on the City of Paris. Do you have the link to the image, hopefully on familysearch? Or provide the transcription - how old was this Michael Ryan in 1892, etc.
One of the men I found arrived on the City of Berlin, and was 23, or born about 1869:
New York Passenger Arrival Lists (Ellis Island), 1892-1924
Name: Michael Ryan
Event Type: Immigration
Event Date: 07 May 1892
Event Place: Ellis Island, New York City, New York, United States
Gender: Male
Age: 23
Nationality: Ireland, Irish
Departure Port: Liverpool
Ship Name: City of Berlin
destination: Illinois
https://www.familysearch.org/ark:/61903/1:1:J6T7-T4W