ADDED NEW QUESTION Best tips for dealing with an arm cast!

Could you wash his hair for him at the sink?

Glad press and seal to cover the cast during showers. It will stick to the cast and skin and we found it works better than a cast cover. We did a couple of layers of press and seal and then a trash bag. Hair is really hard one handed — it’s doable though with a little practice especially if his hairs pretty short. My girls and I all three have had casts over this year — their ankles and my arm — we all found a hand held shower head very helpful to try to keep the water away from the cast. We also used body wash wipes to help clean up between showers because showering was a bit of a production. I even took my oldest to the hair salon just to get her hair washed several times because she had really long hair it was just difficult to really get it clean even with my help. We tried the sink and it didn’t work very well. The wipes are especially helpful for the areas right around the cast. Also keep a good moisturize around for his fingers and hands — they will start peeling and once the cast is off his arm will really peel for a week or so.

ETA: I hope he’s feeling okay. My youngest broke her growth plate at the ankle last year — she said it was pretty painful the first 2-3 weeks but then the pain got significantly better. We rotated ibuprofen and Tylenol but she still had a really hard time at night with the pain for the first couple of weeks. If we can prop his arm at all — it helps with the swelling and pain. I hope everything heals well

In the sink..either he bends forward or sits in a chair. It can be done.


This!

Another thing I just thought of — both girls found it helpful to ice right above the cast and their toes when they were having a lot of swelling. The swelling caused the worst pain for them — one broke her ankle and the other had ligament reconstruction surgery. The surgeon highly recommend icing after surgery but the orthopedic dr for the break oked it as well.

It’s not easy but you can wash your hair with one hand. I had to do it when I broke my arm.  I have something similar to this link is my shower and it was a big help. You can hold your hand under it and pump with your thumb.  If you have a space for it you could put a little shampoo with a bit of water in a small cup that he could pour on his head. A loofah on a long handle makes back washing easier. Put the body wash on it before he gets in then all he has to do it wet it.  

Field Dressing Gloves (like what you would use to clean a deer) from Amazon! Game changer when I was in a cast for broken wrist (cast over the elbow). 

The top of the cast cover we tried was a little different and it leaked. We didn’t have any leakage with the press and seal. If that style of cover worked well before it probably would now as well. The style we tried leaked.

Hand shower and shower chair will help a lot

Also if your kitchen counter is long enough he can lay on his back with his head in the sink to wash his hair, if the other way won’t work. Maybe some dry shampoo for in between washes? My kids were both girls so I don’t know if a boy would go for that.

Nope nope nope. Nothing goes in the cast , nothing. Always use a hair dryer set to cool.

ASOLUTELY NOT
Do not EVER do this.

I'm already assuming the OP has more sense that this but in case anyone else reads this and thinks it might be a good idea, it is always a definite NO and make sure the kid knows this in no uncertain terms.
Firstly something can get stuck down there and either require the cast to be removed, or the person gets too embarrassed to tell anyone and it stays there and causes skin breakdown and infection.
Even something with a blunt edge can cause a scratch to irritated, moist swollen skin under a bandage, and any break in the skin under a plaster that can be observed or cleaned is in danger of getting infected.
I've seen a child need skin graft to their arm because they used a pencil to scratch under the cast and broke the skin. He kept complaining his arm hurt but his parents thought it was from the break and kept giving him pain medication...then it started to smell and when they took the cast off he had quite a large ulcer under there that required a skin graft.

I also wouldn't recommend those gloves or any other contraption that encourages use of his hand, the point is the limb is meant to be resting to allow the bone to heal, finding creative ways to use the hand as much as possible isn't advisable.

I broke my arm multiple times when I was in elementary school, I echo he’ll figure out how to wash himself including his hair - it’s really not that big of a deal. A pump shampoo/soap is easier, but even that isn’t essential. Have him watch the old Amazing Race with Bethany in it - the girl who lost one arm in a shark attack - it’ll give him an appreciation for what one can do with one arm.

I edited my post. I changed it to say, a technique to relieve the itch.