Oh my god...
I was first going to reply with "There is no USB you loon", because I've been using these displays for years as travel monitors and have never seen one despite routinely handling them at all angles when carrying them around. Instead, I did a bit of searching and, lo and behold, the god damn thing has a friggin USB port! A very well hidden USB port... I'm just boggled at why it's hidden away like that. I actually found a few professional review sites that specifically called out the lack of a USB port in their reviews.
Apparently the manual calls it a "service port" or something, but it can be used to charge the glasses and install firmware updates. I'll have to test it tonight to see if it gives enough juice to power the Chromecast. That'll be pretty rad if it does. Also hoping the Chromecast can fit into the HDMI port without the dongle thing that comes with it so it's not flopping around all the time. I'll find out more when I get home and handle the thing a bit.