I couldn't find Jeremiah McCarthy when I searched but to find ships arriving New York from UK & Ireland, I went to Ancestry's New York Passenger Lists 1820 - 1957.
I didn't fill in any name. I just typed in the following:
Arrival: 21 Sep 1885 New York
Departed: UK
Race/Nationality: Irish
Click Search and some ships come up on the list with the names of passengers shown.
For instance, I saw passengers for the "Brittanic" and "State of Alabama" and "Etruria" - but I didn't go further.
To widen your search if you can't find him, you might care to reset the search. Try leaving out the word Irish in the Race/Nationality box. If you think he may have departed from Liverpool, try that in the departure box instead of UK.
_______
It may not be enough to use Jermiah's name and details and look for him on the ships you see arriving on the correct date. Maybe his name was wrongly spelt, or perhaps the ship arrived on the 20th or 22nd instead of the 21st. If his name is wrongly spelt, you may need to look at the pages of the ships passenger lists one by one.
Brian