I love how that cylinder is plumbed with little tiny elbows and unions and stuff. Things are so much easier these days with nylon air brake tubing instead.
So the door is operating by applying a torque on the bar that looks gray or maybe a baby blue color in the fourth picture? The arm must belong in, but is absent from, the third picture.
Not being a pneumatics engineer, but having stayed at Holiday Inn Express once
, I'd suggest doing some experiments. Install the arm on that rod and measure how much force is required to operate the door -- attach one end of a fish scale to the arm and pull the other end by hand. Note the highest reading on the scale. If for example it peaked at 45 pounds, and if your air supply is going to run 90 PSI then as a starting point you might try finding a ram with a half-square-inch cross section to operate it (90 pounds/square inch times 0.5 square inch = 45 pounds).
Alternatively, choose some arbitrary ram and stick it in there. Operate it with shop air at a low pressure and test whether it moves the door with acceptable speed. Adjust air pressure up or down until you like its performance. When you find a good pressure to use on whatever diameter ram you had on hand, you can then re-proportion it to figure out what ram diameter would be best for whatever air pressure your trolley will run. Then go shopping for a ram with the right diameter and travel.