Thank you. It works when I do as you said: Object mode -> Shift select Mesh – Weight paint Mode.
In my head I thought I needed to activate Pose mode first. I’m not sure where I got that idea from.
When switching to weight paint mode, the armature does seem to switch to Pose mode, so maybe that’s why I got confused?
In a different Blend-file, I am able to switch between objects by holding Ctrl-Alt and leftclicking the armature while weightpainting. Then the armature becomes the active object and shows as being in pose mode. I’m not sure why I can do it in that file, but not in the exercisefile.