I go biking every morning. I slide the iPhone in the pouch underneath my saddle, zip up and ride away. I'm thinking of getting a small handle bar pouch that I've seen on other bikes.
The only time I use a cheap phone is if I expect to be in a place where I might fall into water or otherwise get splashed on (kayaking, going to a water park, rollercoaster park etc).
You can swap out the iPhone SIM card. Most GSM phones will accept them. My old Ericsson T28 is 11 years old and it read the info on the iPhone SIM and connected to the network. I was impressed!
Your only problem is that your contacts are stored on the iPhone not the SIM. As far as I know, there's no option to copy your contacts to your SIM. You'll have to do it the old fashioned way (input directly on cheap phone).