First of all, you wouldn't go to Fort Gibson, which was what sat on Ellis Island at that time; it was an active military depot. The processing station only dates from 1892. In New York City, you'd go to lower Manhattan directly. Fort Clinton was the immigration station there, but even that only opened in 1855, after the worst of the famine. Before that, your situation would be quite different, and you'd just go to the port; there was no settled restriction policy or quota system, although the nativists and Know-Nothings were trying to change that in the 1840s and 1850s.

That, however, is only arrival itself. Nobody was granted citizenship at Ellis Island on arrival (or anywhere else) unless, after 1855, the spouse or child of a citizen; the naturalization process in the US had to be followed and that is laid out in the Constitution. The Naturalization Law of 1802 governed that at the time, and it required the recording of all aliens on their arrival in the USA to document that if they were later to apply for citizenship. So you'd register with the Clerk of the Court in Lower Manhattan and get a stamped certificate atttesting to your arrival. As a free person who was ostensibly white (naturalization was not fully colorblind until 1952, despite the inclusion of people of African descent, mostly ex-slaves, after 1870) you had to reside in the US for five years before you could petition for citizenship. You'd have to declare your intent to renounce your existing citizenship and pursue US citizenship a few years prior to that. (That didn't mean your petition would be accepted, or that you wouldn't find other roadblocks.) As far as I recall there was an Oath of Allegiance, but all civics examinations were devised by the judges considering the petitions, usually orally; standardization only came in the 20th century. (The USCIS actually has a fair bit of overview material on the development of immigration; yes, it did get tougher with time.)

Joyce Vialet's old Brief History of US Immigration Policy is of some use here, but William Pencak's edited collection History of Immigration to New York (1991) might have more specific information on some aspects. If anyone studies this closely, they may have other suggestions as well, especially regarding the certain effects of anti-Irish sentiment on this entire procedure. [edit: clarifications]