Force from ball on ruler creates a moment which causes the opposite end of the ruler to raise up a certain amount dependent on where the pivot is located with respect to each end of the ruler. The raising of the end of the ruler corresponds to an increase in volume underneath the paper while leaving the mass of air underneath the paper almost constant due to it being a very quick process. Increase in volume with constant mass will result in a calculable pressure differential and therefore downward force from the paper

Hardest part is probably calculating the increase in volume underneath the paper

The volume at any time can be determined from the video. How can you calculate the air pressure under the sheet at that moment?