How to repair the arms on an upholstered chair?

It depends how broken it is. You need to find first which system they've used to attach it then when you have that you can see how to replace

You could carefully glue on a “patch” with fabric that complements the fabric already there. Or if it’s a solid maybe some embroidery to stitch it together plus a bonus of beautiful stitchery?

The “patch” I’d cover either the entire arm, or just the ends, but all the way from into the cushion area up over the arm and down th outer side.

Here’s some photos, that might help. One I scribbled on to show the location of the partial “patch”. Instead of glueing, you could also hand stitch with upholstery thread and needles and then stitch down a upholstery decorative ribbon to cover raw edge.

Almost forgot photos.

Make arm rest covers that slip on.

Find a similar or coordinating fabric and make removeable covers. Use the little twist pins to hold them on. That way you have the extra benefit of being able to wash them if needed.

I had the same problem. I bought a coordinating fabric and removed the back panel which I used to reupholster the arms and used the new fabric for the back, I also made arm covers and a lumbar pillow for the chair in the same fabric I used for the back of the chair.