Dear Mr. Paananen,
Thank you for posting your request on History Hub!
We suggest that you review these Genealogy: Passenger Arrival List Research Tips and NARA’s page on Immigration Records for suggestions on how to locate immigration information for your grandfather. There are numerous ways that individuals could enter the United States from Europe such as on a ship that arrived at various coastal ports or over land from Canada, etc.
We searched the National Archives Catalog and and 9 series in the Records of the Immigration and Naturalization Service (Record Group 85) that contain passenger lists for the 1880s. For more information, please contact the National Archives at Washington, DC - Textual Reference (RDT1) at firstname.lastname@example.org.
We also located 26 series in the Records of the U.S. Customs Service (Record Group 36) that pertain to passenger lists for the 1880s. For access to these series, please email the reference unit listed as the contact section of the Catalog description.
Due to the COVID-19 pandemic and pursuant to guidance received from the Office of Management and Budget (OMB), NARA has adjusted its normal operations to balance the need of completing its mission-critical work while also adhering to the recommended social distancing for the safety of NARA staff. As a result of this re-prioritization of activities, you may experience a delay in receiving an initial acknowledgement as well as a substantive response to your reference request from RDT1 and various other NARA reference units. We apologize for this inconvenience and appreciate your understanding and patience.
In addition, we suggest that you search local newspaper archives and city directories for mention of Antii (Anders) Paananen during the timeframe that he resided in the United States. Many of the passenger lists, newspaper archives and city directories have been digitized and are name-searchable online using Ancestry and FamilySearch. There may be a fee for using Ancestry. Instead, you may wish to contact your local library. Many library systems subscribe to these sites, making them free for patrons.
We hope this is helpful. Best of luck with your family research!