Um no. It starts the download once you have turned the 360 on since it will already be in your download list but it doesn't wake up the 360 from standby to initiate the download.
Neither the PS3 nor the 360 had this feature. Only the PS4 has for now.
Yes, if the 360 was on it would start the download. XB1 is always on and even if it weren't, it does not start the download automatically when the XB1 is on so it is an old 360 feature that is missing.