The contacts aren't stored on the SIM. You can back up your contacts in an email account and then sync them through iTunes. Another way would be to restore the old phone from a backup of the iPhone 4. That will put contacts on the old one, but backing them up in Google is still a good idea.
Your SIM will work fine in the old iPhone, but you'll need to buy or make a SIM adapter since the iPhone 4 uses a micro SIM and the older iPhones use the standard size. Another option would be to get a new standard size SIM from AT&T. I think they are $10, whereas you can make an adapter for free using one of your old deactivated SIM cards.