The 6 on the left of the photo shows you that it was the sixth race. Some of the people in the photo are wearing kitty hats, and if you speak to one of the men in the ticket booth (the grouchy one, I think), he'll tell you about kitty hat Tuesday. That gives you the day. To find out the week, you need to know which cats are racing. If you speak to Maximino, he tells you about the first time he proposed to Olivia, when her favourite cat, Sanspoof, was killed. In the photo, you can see Max's proposal on the side of the blimp, so it must have been the race during which Sanspoof was killed. If you go to the hallway outside the kitty litter room, there is a shrine to Sanspoof. Read the plaque, and it tells you the week of that particular race. That gives you the three pieces of information that you need.
The way I've written it makes it sound a bit long winded, but it can be worked out