Don't know about hyperextended joints specifically, but the general rule for musculoskeletal injuries is rest, ice, elevation, and compression.
You want to keep icing until it doesn't swell anymore; hot packs increase circulation which helps in healing (later on) but also increase swelling.
I get a lot of leg pains after running, and ice and elevation seem to work best. I generally lie down with my butt in a corner and my legs extended up a wall a couple times a day for 15 min or so, when I'm reading or something. Maybe try something like that with your arm to reduce swelling in the joint.
Still, seeing a doctor is probably a good idea to make sure there's no ligament damage.