About all that you can try is to "export" or save a copy of the configuration file for the AirPort Extreme and then see if there might be some way to "import" those settings from the new router.
Open AirPort Utility
Click on the AirPort Extreme, then click on Edit in the smaller window that appears
Click on the File menu......up at the very top of the Mac's screen where you see other menus like Help, Edit, etc
Click on Export Configuration File and choose a destination for the file, like the desktop, etc.
Realistically, the chances that this will work on a non-Apple branded router are a tiny fraction above zero, so keep your expectations low.
Go ahead and take a series of screen shots for each of the devices that require the special port settings, so you can do things the old fashioned manual way if the new router does not have an "import" feature. Off hand, I can't think of any that do, but you never know.