My son as a youth always iced his arm, when he went to college the pcoach told us that he lets his pitchers and catchers do what they feel comfortable with so with that as a starter, he continued to ice his arm until about a few years ago (reliever). He will ice his arm if he feels a bit sore but he doesnt feel that it helps one way or the other. There are many that say its essential and many say unless you have an issue its not necessary and there are some pro/college teams who do not ice their pitchers after outings.
I think that icing young arms is less as important as making sure that they are not over used. Perhaps not letting him play a position so soon (especially one that requires less throws) might be more beneficial.
JMO