Yes, you can request a continuance on the day of your hearing. However, it will the court's decision whether or not to allow it. You should request a continuance sooner if possible to give everyone as much notice as possible.
It sounds like just getting a continuance, however, will not resolve the issues you are facing. Normally, a person needs to show a "significant change in circumstances" to modify custody or support. In order to answer your questions, an attorney would need much more information and to throughly review your case. I suggest you seek legal counsel on your issues.
THIS ANSWER IS FOR GENERAL INFORMATIONAL PURPOSES AND DOES NOT CONSTITUTE LEGAL ADVICE. IT IS BASED ONLY ON THE LIMITED INFORMATION AVAILABLE. THIS ANSWER DOES NOT ESTABLISH AN ATTORNEY-CLIENT RELATIONSHIP.
Yes you can always ask for a continuance in a child custody matter if you have a good reason to continue ie. I need more time to review the moving papers. You are correct in believing that he won't get custody. However, the law does state both parents should get frequent and continuing contact with their children. However, in your case since he has been gone for two years the court will require he give you information where he lives and his phone number. The visitation will be quite small at first maybe even supervised.