Whilst you can provide opportunities to explore feelings of empathy, you can't teach it as such - if the emotion is absent for example. It's something we develop over time, alongside our other emotions. Children learn at different stages, there's no one size fits all, but a 13yr old unable to empathise is something I feel needs to be looked into by a professional to explore why.
If however your DS is able to recognise emotional distress in others, but not care, then that's another different issue that very much needs professional intervention.
In school we explore emotions, how to recognise our feelings, how to act appropriately to them. For example, it's OK to feel angry, it's how you act on it that matters. Lots of children (I'm talking year 6 here, so 11 year olds) still struggle with identifying their feelings and then how to handle them. They look to adults to solve their problems for them, rather than develop skills (supported by adults) to help themselves.
So whilst there are some things that can be taught, if the feeling isn't there in the first place, you've got nothing to work with.
Some links that may help:
www.psychologytoday.com/gb/blog/smart-parenting-smarter-kids/201905/how-children-develop-empathy
www.parentkind.org.uk/blog/8719/Helping-children-develop-emotional-literacy
Whilst looking into the above, and accepting that you have some hard work ahead of you as a parent, I would also be contacting the school to see what consequences they are putting in place as it happened at school. They will have spoken to the child involved, the parents of that child, and I would very much hope would follow their policy on consequences.
I would ban SM at present as that seems to be exacerbating the issue and it does not seem your DS currently has the emotional ability to handle being on it.