The website (and Rick) say that one cannot visit the cathedral during masses. Website lists mass times for Sunday in June at 7:00, 8, 9, 10, 12, 17, 18 and 19:00. Would it be safe to assume we can go in between 11:00 and 12, and/or 13:00-17:00? (Trying to get this visit to line up with train time.)
We attended church on a Sunday at Koln Dom. There are minders who keep the tourists near the back of the church while the faithful may go forward and participate, quietly observe or enjoy the music of the mass.
If you want to climb the steeple you need more than an hour. So the time slot between 1300 and 1700 would cater to that. However, the cathedral is located in the North-Eastern corner of the old town. If you want to take pictures of downtown Cologne it might be better to do the climb in the morning when the sun is still in the East.