1. Start any game (except tutorial or modded) between January 1 and January 25. System time is checked, so by changing the date on your computer you can get the achievement anytime. Additionally, as long as the game is started between those dates, the quest should pop up even after the 25th.
2. Play until turn 40 (fast speed) or turn 80 (normal speed), preferably with a fast combat fleet ready. You should the quest Stranger at the Door. Attack the quest objective enemy fleet.
3. Complete the Living Memory quest by colonizing one of the quest objective planets.
The achievement should unlock.