Here is what I think is going on. It sounds like someone installed Mountain Lion on that Mac and you did a command-r boot to the Mountain Lion recovery partition to reinstall. When you get to the part where it is ready to download the OS it prompts you for an AppleID. You will need to enter an AppleID that was used to purchase Mountain Lion, otherwise the download request will be rejected. You have a new AppleID with no Mountain Lion purchase attached to it, so this will not work.
Did you already use Disk Utility to erase the OS? If not, just boot back to the Mountain Lion OS. Then go
here and purchase Mountain Lion. Apple will email you a code you can use with the App Store app to purchase Mountain Lion. Then with that Mountain Lion purchase associated with your AppleID the download will be accepted from the recovery process.
If you have already erased Mountain Lion, you are in a bit of a spot. You will either need to order a Snow Leopard DVD to reinstall then update to Mountain Lion from there. Or you can use a friends Mac to make the Mountain Lion purchase in the App Store like I described.