I can't speak for your specific captain seat, but with my seats pictured below, the arm rests flip up and down as yours do. To remove an arm rest, I need to apply a lot of force inward while rotating the armrest beyond the top-stop continuing rotation backward. When the arm rest is 180 degrees from the normal down position, it detaches.
Your arm rest being broken, maybe you can remove the piece inside with my post in-mind.
Now I read deeper and think I misunderstood you. You simply need to reattach the metal post back inside the armrest. I would definitely remove the armrest cover, break it down to 100% steel and get that pin welded back on. There is a massive amount of force on an arm rest when someone uses it to push off from to get out of the seat. Anything short of a good weld it is futile.
The vinyl cover is usually zippered on, and from your picture I can see the zipper. Unzip it and peel it off. The foam inside might just drop away leaving you with the steel frame. The welding process is extremely hot. Be sure you have ONLY steel on such a small metal frame when welding. Hard plastic and padding could deform, maybe melt and drip off, or even burn off.
