Some of the passenger lists for arrivals in the Port of New York found in Ancestry's database New York, Passenger Lists, 1820-1957 come from original records on NARA's Microfilm Publication M237, which Ancestry cites as follows:
Passenger Lists of Vessels Arriving at New York, New York, 1820-1897. Microfilm Publication M237, 675 rolls. NAI: 6256867. Records of the U.S. Customs Service, Record Group 36. National Archives at Washington, D.C.
These are not immigration passenger lists of the type we see from Ellis Island -- those lists from Castle Garden were stored at Ellis Island and were destroyed in the fire at Ellis Island on June 15, 1897. These are US Customs Service records, which have much less information.
A sample header from the Customs House lists is below. This is from a May 1879 arrival of the ship MAIN at the port of New York:
The departure port of Bremen is listed, but unlike the immigration lists, there is no date of departure.
Departure lists from Bremen for this year do not survive. Die Maus says:
Unfortunately, all lists from 1875 - 1908 older than 3 years were destroyed due to lack of space in the Bremen Archives.
The site Norway Heritage's page for Norddeutscher Lloyd - 1879 says that the Main's authorization and route was
Via Fredrikshavn and Bremen to New York or Baltimore and on to the final destination
However, the only two search results in their database for 1879 are from November, which is after this New York arrival.
FamilySearch's Historical Records Collection Germany, Bremen Passenger Departure Lists (FamilySearch Historical Records) starts in 1904 and is too late for this question.
I plan to search newspapers to see if I can find out any other information about this voyage of the Main, and to review the resources in the answers to the question Finding alternate indexes or record sets for passenger lists / ship manifests from Germany?, but I would like to find other resources that would tell me more about the ship's transit time and the departure from Bremen.
A description of the Main from The Ships List says the MAIN could travel at a speed of 13 knots:
MAIN 1868 The first MAIN was a 3,087 gross ton ship, built by Caird & Co, Greenock in 1868 for Norddeutscher Lloyd [North German Lloyd] of Bremen. Her details were - length 332ft x beam 40ft, clipper stem, one funnel, two masts, iron construction, single screw and a speed of 13 knots. There was passenger accommodation for 70-1st, 100-2nd and 600-3rd class. Launched on 22/8/1868, she sailed from Bremen on her maiden voyage to Southampton and New York on 28/11/1868. In 1878 her engines were compounded by the builders and on 6/3/1890 she commenced her last Bremen - New York voyage. On 6/3/1890 she started her final Bremen - Baltimore crossing and the following year was sold to British owners. She was destroyed by fire at Fayal, Azores on 23/3/1892. [North Atlantic Seaway by N.R.P.Bonsor, vol.2, p.546]