The first or left part of the SpinBox is a LineEdit object so you need to add the Class LineEdit to your theme too. Then you can change everything regarding theme and stylebox for it.
You can set everything in the inspector.
The theme can be set in GDScript too.
But it seems to be missing to set the icon in GDScript or I did not find it yet.
does not seem to work. Although it works to use the same png file in the Inspector for defining the icon.