Today's sun points to Milky Way center.

[Anomalocaris]

Precession of equinox changed the orientation of the earth’s spin axis in space and by extension the obliquity between earth spin axis and the ecliptic plane.     But it does not change the orientation of the ecliptic plane in space.    It is the  orientation of the ecliptic plane in space that determines when, during the sun’s 240 million year orbit around the Milky Way, could the earth, sun and center of the galaxy come close to being arranged in a straight line during any part of earth’s orbit around the sun.    

Basically that happens twice during each of the sun’s 240 million orbit around the galaxy.    The closest angular approach the sun appears to make to the center of the galaxy during its annual passage around the ecliptic changes by about 1.5 degrees each million years.   So in about 120 million years,  it would move 180 degrees and the sun will transit the general region of sky that contain the center of galaxy in June.    Between roughly now and roughly 60 million years from now, the sun will move 1.5 degrees further from center of Milky Way during December each million years.   In the 60 million years after that, the sun will move closer to the center of Milky Way during June by 1.5 degrees each million years.