You can buy relatively inexpensive "outlet" timers. I use one when on vacation to turn on and off at various times during the day and night lights and loud stereo.
I use an analog timer, but I am sure there are digital times if you need accuracy.
I should add that if you need a source that will play automatically you probably will need a CD player or FM tuner as input to the WIIM. I don't think a WIIM will stream automatically on startup.