Princess will assign you a room when it's convenient for them.  TBH, the night before embarkation is the latest I've ever been assigned a cabin.  I make it a habit to re-check my Travel Summary the night before a cruise to make sure that I didn't receive a upgrade (one can hope).  Since I arrive at least a night before embarkation, I always bring a laptop so I can do this and print my luggage tags (either by accessing the hotel's business center or forwarding PDFs of the luggage tag to the hotel