Just swap the sim card into the new phone.
I believe the GS3 is a MicroSIM, and the Captivate is a "regular" SIM. So the person can get a MicroSIM from AT&T. If you call for one, I think they charge you $15. But in store, they will often give it to you for free to AT&T account holders.
You can also cut the SIM to make it work, using guides online. But its probably easier to just get a free new one from the store.
My contract just ended and I have the $30 unlimited data grandfathered. I've always been able to keep it when upgrading by speaking to customer service. But this time I don't want to upgrade for another 2yr contract. I've been a customer forever and would like to have the freedom of no contract for a little bit. I'm currently using a 3g samsung captivate but want to get the s3. I found a really good deal on craigslist. Will at&t let me keep unlimited data without upgrading? I'm just worried that they tell me yes when I call, then after I buy the phone they say no you gotta upgrade. That would suck..
Dont' worry, they can't make you upgrade, you can use whatever hardware you want (as long as its compatible with their network).
If you don't presently have LTE on your account, and the phone you are buying is compatible with AT&T's LTE bands, you can get it added to your account w/o an upgrade and w/o a new contract. But you have to go through AT&T to do it (either by phone or in an AT&T store). Simply tell them you want to keep unlimited data, and add LTE. They technically have to change you to a "new" data plan that is unlimited data with LTE, but its really just the same - unlimited data, but with LTE capability added and no extra fee (its just what they do to make it work in their system). Just make sure they do it right before you leave the store or hangup the phone. Its not completely straightforward, it took them about 15 minutes to get it right for me at my AT&T store, and the rep had to bring the manager into it, and they were
really trying. But its worth the hassle, since if they screw it up and take unlimited data off your account, its a real pain to add it back.
And this is the only way to add LTE to your account, if the S3 you are buying has LTE.