Now, let's say someone was thinking about doing a shareware game, how would they handle customer support?

usually via email, a support form on their web site or a forum (better because other users can offer help - worse because any problems can get moaned about in public). Some developers offer customer support by phone others don't. High price shareware (biz apps not games) usually offer a phone service because business customers expect it. Other options to consider would be to offer a paid customer support option or a premium rate support line.