I am struck by the difficulty of providing a straightforward answer to what would seem to be an obvious and significant question.

Compiling information from several different sources, however, the essential details and statistics are as follows:

1. The first line connecting Chicago to New York, via Buffalo, was opened on 22 February 1852, although this certainly does not mean that one could board a train in New York and sit on it until it pulled into Chicago. The journey at this time incorporated numerous changes of train, together with a gauge break and a ferry crossing of the Hudson River.
2. The earliest direct railroad route would have run from Chicago to Detroit, Detroit to Cleveland, Cleveland to Buffalo, and then Buffalo to New York.
3. By 1854 three railroads offered services between Chicago and points east: the Michigan Central, the MSNI, or Michigan Southern & North Indiana (both of which reached the city in 1852 and provided services to Detroit and to Cleveland and Toledo), and what became known as the Chicago & North Western (1854).
4. The New York Central railroad, which incorporated in 1853 and was by 1856 the largest business corporation in the US in terms of its capitalisation and annual revenues, made deals with these companies to offer the first scheduled services running between New York and Chicago.
5. By 1855, when the Chicago Branch of the Illinois Central Railroad was completed, 58 passenger trains were arriving and departing the city daily, and Chicago had become the focus of ten trunk lines with 2,933 miles of track leading from it to all parts of the country.
6. William Cronon, in his Nature's Metropolis: Chicago and the Great West, says that the journey from New York to Chicago took about two weeks in c.1850, and adds that with the arrival of the railroad "it took less than two days".
7. He provides a helpful map, drawn from the 1932 Atlas of the Historical Geography of the United States, to show the speed at which it was possible to travel westward from New York in 1830, as compared to 1857, which you can see here.

A clearer idea of exactly what was involved here can be had from the Annual Report of the Board of Directors of the Michigan Central Railroad (1847), issued at a time when the route was being planned and built. This contains a table which estimates that the rail journey from Detroit to New York (672 miles) would take 24 hours, and that between Chicago and New York (951 miles) would take 35 hours. This estimate, it should be made clear, is based on average speeds ("express trains are figured at 33 miles per hour, on double tracks, and 28 miles on single tracks, including wood and water stops") and not on an actual journey from point-to-point, including waits for connections. If changes are added to the figures, a total journey time of a little less than 48 hours seems reasonable, and hence Cronon's "less than two-day estimate" looks accurate.

The same sort of speeds were emphatically not available for freight traffic at this early date, which is not surprising when it is borne in mind that each switch of trains and gauge would have involved some complex shunting and/or transhipment of tons of goods. By the late 1850s, a "fast" Chicago to New York freight service was offering journey times of eight days at a cost of $1.75 per hundred pounds of goods.

Passenger travel time had reduced to 24 hours by 1881 and 20 hours by 1893.

We can conclude from the difficulty of arriving at a precise figure for travel time in the early 1850s that rail-only journeys all the way from Chicago to New York probably were not made very often in the early years. There is a good reason why the railroad route was not favoured by travellers at this early date: steamboat transport via sidewheel/paddlewheel boats on the Great Lakes between either Detroit and Cleveland or Monroe and Buffalo was more comfortable and more popular for quite a few years. Poupard's book transcribes an 1853 advertisement in the Monroe Commercial newspaper that points out it was also faster to break the rail journey by taking a steamship across Lake Erie from Monroe to Buffalo than to complete the equivalent journey by rail.

Sources

"Early Chicago, 1833-1871. A collection of documents from the Illinois State Archives"

Annual Report of the Board of Directors of the Michigan Central Railroad (1847)

A. T. Andreas, History of Chicago. From the Earliest Period to the Present Time (1885)

Albert J. Churella, The Pennsylvania Railroad, Volume 1: Building an Empire, 1846-1917 (2013)

William Cronon, Nature's Metropolis: Chicago and the Great West (1991)

Gary Gorton, Ante-Bellum Transport Indices (1989)

John F MCDonald, Chicago: An Economic History (2016)

Bessie Louise Pierce, A History of Chicago, Volume II: From Town to City 1848-1871 (1940)

Marcel Poupard, Railroad Articles from Monroe County, Michigan: 1836 to 1965 (2009)

William Ganson Rose, Cleveland: The Making of a City (1990)

Ronald Seavoy, An Economic History of the United States: From 1607 to the Present (2006)

John H. White Jr., Wet Britches and Muddy Boots: A History of Travel in Victorian America (2013)