you can freeze wet towel as a cheapo reusable option but i think cold compress is not exactly the same as it is prolonged (still worth a try!), compared to shower.

it's not full body cold shower, just neck-up, so can be done even in the sink, if your faucet is high enough (although if you have long hair definitely too much hassle). i think the theory was some vessels might relax after cold shock in the neck and skin, and help circulation or something, dunno if it's some belief or real shit tbh.

It kinda feels like head goes numb from cold, and then sometimes with numbness headache disappears (and sometimes doesn't, but still pog when it works, takes like 10 minutes to determine what happened and go to drugs if it didn't)