Is it okay for the arch support on my new running shoes to hurt my foot?
It’s not ok for your feet to hurt and most likely it won’t go away either. Running shoes are designed for different foot/bone structures. The brand or shoe name doesn’t tell much, so you need to do some research yourself or go into a specialty store. You might have to spend a bit more money there but if you talk to a trained sales person, you will learn a lot about the type of shoes that you need and which ones you should avoid. They might have you run on a treat mill to see your gait and look at your bare foot to see whether you need little or lot of arch support. Then they show you a few different options (brand/designs) that are tailored towards your foot structure. I have some great XC running shoes (popular brand and not cheap) but I can’t wear them for anything else but “running errants” as I get blisters after 2 miles into a run. Here is a pretty neat running shoe advisor from Brooks (for their brand shoes):
