When Castle asks the computer voice "where are we," the voice answers "You are aboard the vessel Tenzing Norgay..." Tenzing Norgay was the first Sherpa to summit Mount Everest with Sir Edmund Hillary on 29 May, 1953.
Early in the story, they run a "trace of IP addresses" while trying to identify a computer hacker. None of the addresses shown are possible. IP Version 4 addresses are in 4 groups of 1-3 digits each, not 5, and no group can be larger than 255. An address such as 289.462.938.17.4 is nonsense. However, this isn't a "goof". It's the modern equivalent of using prefix 555 for all phone numbers in movies and TV - intentionally done to keep people from bothering any real Internet sites.
Instead of the usual bridge/NYC city skyline with a pen as part of the Castle logo, the pen is replaced with the motorized screwdriver that was the murder weapon.
Castle puts special emphasis on the word "destiny" which is likely a reference to his voice acting in the game Destiny which partially takes place on Mars.