Since the question says that the tourist visited 5 attractions by taking the shortest route, it makes sense to add the shortest routes possible from the hotel.
So they definitely went to the Courts which is 60m.
They then took the shortest route to the Fountain, which is 80m.
Then the Arch which is 80m from the Fountain.
Then the Castle which is 90m from the Arch.
Then the Tower which is 110m from the Castle.
Then back to the Hotel which is 110m from the Tower.
Adding this up: 60 + 80 + 80 + 90 + 110 + 110 = 530m
So the tourist did not go to the Palace.