if your Mac's serial number did not change, you may be able to run Aperture by booting from the old drive. If it launches, look at the "About" panel:
From Aperture's man menu bar use the Application menu "Aperture > About Aperture". This will display the registration information and Serial Number.
P.S. How did you originally buy Aperture? As a boxed retail version? The serial number should be in or on the box with the installer CD, on the booklet inside. If you bought Aperture from the App Store, no Serial number should be required.
if you cannot retrieve the serial number, contact Apple: Pro Application Replacement Serial Numbers