Awww, poor baby. My daughter broke her arm last summer right after she turned 3yo, and I thought that was rough... At least she could still walk around.
For getting around, I think I would borrow/buy a wagon if you don't already have one. Then you can take him around in it. Maybe the novelty of riding in the wagon will help him get over the frustration of not being able to walk. Or just use a stroller... But a wagon would be more fun. ;)
For getting around on his own, I would almost be tempted to get him a skateboard or scooter, and let him sit/lay on it to scoot himself around. Just make sure he keeps his hands wide to avoid rolling over his fingers.
If you have a smartphone/tablet, mayb you can download some apps for him to play on. Take him to the library for story time, and get a bunch of picture books for him to look at. Heck, you could even play at e library for a while. Most have a children's section that has a play area where kids can read, work on puzzles, do puppets, etc. Even if he is doing the same ting he would do at home, the change of scenery will help keep him stimulated.
Use the wagon, and take him on walks. You can take him to the zoo, or museum, etc.
You could make yourself a little sandbox. Just get a cheap plastic pool, and fill it with a few bags of play sand and some sand toys. (Just make sure to wrap his cast really well so he doesn't get any sand in it.)
I would also browse Pinterest, and search 'sensory bins'. They are a great sit-down activity that keep kids occupied well.
I would look online and order a cast protector for bathing. I used plastic bags for my DD, but 5 weeks is a pretty long time so I think it would be worth the investment. I would also pick up some moleskin, and use it in places where the cast tends to rub more to help protect his skin from chafing. The padding does compress and move around a little bit, so even of you don't need it right now you may need it later. You can also use it for areas where the cast seems to get grimy... Put the moleskin on the cast, then just peel off and throw away as needed. :)
I hope he heals quickly.