Can Rubella virus pass into brain stem?
It obviously can do that, causing a rubella encephalitis. A specialized form seems to be the "Progressive rubella panencephalitis", which is very rare with only 20 cases described since its first description in 1974.

About rubella encephalitis more papers are available, see these for a start:


Neurological aspects of rubella virus infection.
Mumps encephalitis: return with a vengeance.
Neurological Aspects of Rubella Virus Infection