Before October 1, 1874, Jews in Germany had to report their births, marriages and deaths to the competent court, where these were registered. Unlike many records of Jewish communities that were destroyed, many files exist here. Here are lists of Jewish births in the city of Königsberg in 1859. The scanned originals can be viewed at www.familysearch.org
Pdf list of 5 pages with 80 births, incl. the names and professions of their parents.

Link to Jewish Births 1859

One more side note: the municipal judge in charge at the time was Karl Eduard Güterbock (1830-1914), who had converted from Judaism. He later taught as a professor at the Albertina University of Königsberg.