We purchased tickets online for our own self-guided tour of Sainte-Chapelle as soon as we arrived because we had not pre-booked. It would not have mattered anyway because the queues are long and very confusing. If you arrive without a ticket, IMMEDIATELY go online and purchase a ticket for a specified entry time, especially if you’ve purchased the Paris City Pass. There are lines for those with tickets showing entry on the hour: 10, 11, 12, 1, etc. and a separate line for those with tickets showing entry on the half-hour 9:30, 10:30, 11:30, etc. There were many confused people jogging from one line to the next because there is no clear direction. That being said, DO NOT MISS THIS TOUR!
The building is absolutely spectacular, the architecture stunning, the stained glass is absolutely breathtaking and the combined effect is absolutely transporting.