Hi,
You may be in luck because for the dates you want (1885-1889) there are indexes to ships passenger lists for most ports.
You can get a list of film numbers (NARA & FHC) at
http://olivetreegenealogy.com/ships/nara_indexbyport.shtmlScroll down to the state you want and click on the link, then find the film # you need, then decide whether to order it in to a nearby FHC or have NARA do a lookup.
The staff of the National Archives will undertake a search of the original records for a fee.
To find your nearest FHC you can go to
http://www.familysearch.org/Eng/Library/FHC/frameset_fhc.aspYou might also want to try naturalization records, if you need to narrow your time span for looking. There is an explanation and Resource Guide to Naturalization Records at
http://naturalizationrecords.com/ There are also many links to searchable online naturalization records at
http://naturalizationrecords.com/usa/Choose your state of interest from the list at the bottom of the page then click on ONLINE records.
Someone else suggested census, and that's a great idea. The 1900, 1910 ,1920 and 1930 census identify citizenship status, with notations showing the individual was an Alien,
or had started the Naturalization process or had his final papers. They also give the year of immigration.
As well, 1920 gives the year of naturalization. I'll have a look in some of my sources, see what turns up as well.
Almost forgot, New York is the one port that isn't indexed on film for the years you want. You can search in Castle Garden lists - I'd use STeve Morse's one step search engines at
http://www.stevemorse.org/ He has TWO for Castle Garden, one searches the free CastleGarden.org website, the other searches Ancestry's Castle Garden database. CG.org is not complete and it has other ports of arrival mixed in with NY (but not designated as such); Ancestry is complete and only has NY lists