For an Arduino and ESP8266, can I power up the ESP8266 from a set of 2n2222 transistors?
Make your life easy.  Read the datasheet for the ESP8266.  

Page 27 tells you to use CHIP_PD to turn it on and off.  

Page 12 lists all the pins and describes them. There's no CHIP_PD listed, but CHIP_EN is there with the same description.

When it says that pin sends it to low power mode, it really means it.  Setting that pin to disable the ESP8266 drops current draw down to 0.5 microamperes.  That ought to do ya.



Also note, this suggestion was also made in an answer to the linked question.

Doing it the way it was designed to be done saves you a transistor and debugging if you conect it wrong.