I just checked my copy of Winifred Lovering Holman, DESCENDANTS OF ANDREW EVEREST OF YORK, MAINE (1955) under SCHULTZ(E) and variants as well as EVEREST, Sarah Harriet/Harriet Sarah, and am sorry to report that I found no matches under any. This book traces the descendants of a 1640's immigrant (the extremely obscure Andrew Everest) to York, Maine (few of whose descendants were to be found in NYC by the 19th century). Most, but not all, American Everests are descended from this Andy and his even more obscure wife Barbara ---- (my great-grandmother was, but due to sloppy correspondents in the 1890s, doesn't appear in the genealogy under her dad's entry). Most other Everests in this country are descended from later immigrants from England, as noted by several posters to this Forum. Sometimes, too, immigrants from other countries took English names that sort-of sounded like their original surnames.
If you know the state where Sarah Harriet (Everest) Schultze died, I think you would do well to order the death record if you have not done so already. By 1936 recordkeepers had long since been required to obtain the decedent's parents' names and birthplaces, where known, from family or other informants.
Also, check the 1880 soundex for Sarah, probably in New York City if you believe that is where she was from (the 1880 soundex lists only families containing children 10 or under). She should appear also in the 1900, 1910, and 1920 censuses under whatever name she was known by. 1900 and 1920 are soundexed. 1930 is supposed to be released next year, in 2002. Since 1850 the census has asked for state or country of birth for each person, and in 1880, 1900, 1910 and 1920 there have been columns for state/country of birth for father, and for mother. The 1900 and subsequent censuses inquire about when immigrants came to the US.
Hope this is of help, and good luck with your search!