When West first arrives at Ft. Alcatraz a big pine forest can be seen in the distance beyond the walls. Alcatraz is a small island in San Francisco Bay, and the only native plants on the island would have been shrubs and succulents which needed little watering.
The opening scene has West and Pike in a Chinese restaurant, opening and reading fortune cookies. Fortune cookies actually originated in Japan, and were introduced to the US at Golden Gate Park's Japanese Tea Garden restaurant sometime in the 1890s. Until around WWII, they were known as "fortune tea cakes". At about the same time as the name change, the dessert changed from being a characteristically Japanese item to a Chinese one, possibly as a result of wartime politics.
The message from Artemus Gordon is recorded on a disc-shaped phonograph record. Since the time frame of this series is the presidential term of Ulysses Grant (4 March 1869 - 4 March 1877) this causes two problems: first, the phonograph was introduced by Edison on 21 November 1877; second, the disc-shaped record was not introduced until 1888, leaving both elements of this technology slightly in the future for West, Gordon and Pike. This means that the cylinder-based dictation machine seen in other episodes is equally anachronistic. (As a side issue, the term "gramophone" and "talking machine" were early common names for the same device until early the 20th century, the latter being most commonly applied to Edison's cylinder-based devices.)
Near the end, while preparing to launch a rocket, a countdown ("5...4...3...2...1") is heard. The countdown was introduced as a dramatic device by director Fritz Lang in his 1929 film "Frau im Mond" ("Woman in the Moon". Until then, the common practice had been to count up to a predetermined number (usually five or ten), but Lang felt that counting down to zero, where the endpoint of the count was known, had a greater dramatic effect. When counting down (or up) in English for firing weapons and explosives, it is common to omit "five" because it sounds too much like "fire".
Alcatraz was decommissioned as a fort in 1866, when it became a full-time military prison. It would not have been correct in the 1870s to continue referring to it as "Fort Alcatraz" (though the name might have lingered through force of habit).