"I can't seem to find an equivalent in Arduino for the set_time() and time() functions."
That's because they don't exist. The Arduino has a concept of how many milliseconds or microseconds have elapsed since booting so I suppose that you could write your own functions, but the Arduino is limited to 32 bits of milliseconds so you would have to deal with THAT limitation also.
However, the usual uses of Arduinos only require millis() or micros(), so Unix time is usually unnecessary.