What does the phrase "he's pulling an Amaterasu-sama on us" mean? Who is that Amaterasu character they're referring to in episode 6?


  "He always goes straight to his room when he comes home. He's pulling an Amaterasu-sama on us."
This is probably a reference to one of the most famous episodes involving Amaterasu, the sun deity of Shinto myth. 

In the beginning, the world (or maybe just Japan) was created by Izanagi and Izanami. As a byproduct of this, the three sibling gods Amaterasu, Susanoo, and Tsukuyomi came into being (the latter two having domain over the seas and the moon, respectively). 

Amaterasu and Susanoo had a long-standing rivalry/conflict. As part of this, one day, Susanoo threw a flayed (i.e. skinless) horse at Amaterasu and performed some other crude acts. Amaterasu, very angered by this, entered the Ama-no-Iwato, a cave. There she remained for a very long time. During this time, it is said that the sun did not rise, and this was widely considered to be a bad thing. 



As such, the character being referred to in your quote is displaying Amaterasu-like-behavior - holing himself up in some secluded place, just like Amaterasu did when Susanoo angered her. 

Addendum: this is a very well-known tale in Japan, and probably would have been about as obvious to Japanese viewers as a well-known Biblical parable would have been to American audiences (or something like that).