Having found James McAneny's death record, I was inspired to keep going on the family. Here is his brother Michael's naturalization card. As far as I can tell, this was some kind of summary of information made after the fact, along with the immigrant's current address.
Michael McAneny Naturalization Card-front
Michael McAneny Naturalization Card-back
Michael McAneny, Ward 21, precinct D
present address: 1260 East 80th St
Office No. 32876
Native of Ireland, naturalized July 10th, 1901
in U.S. Southern District. Court, New York, New York County, NY
age when naturalized: 29. age when recorded: 42.
1230 E 84th, on 20 October 1917, Ward 21, precinct F
1342 E. 94th, on 23 October 1919, Ward 21, precinct M
So Michael was naturalized in New York City about 1901. I wonder if the family settled there when they arrived. I found James in a Cleveland City Directory in 1898 along with someone that might have been their father. And I didn't find any other immigration records for the McAnenys in Cuyahoga County. Of course that doesn't mean they're not there.
Image of naturalization index card from Ohio County Naturalizations at Familysearch.org