I believe there are some limitations on the lighting controls for the Wraith Prism if you use the 4-pin RGB cable. Namely, you can control the fan's color, but not that of the outer ring or AMD logo. For that, you need the USB cable.
My motherboard only has 2 USB headers and they're both taken, so that's not an option.
The cooler's USB adapter connects to a USB 2.0 header, though your motherboard probably has at least one of the larger 3.0 headers as well, unless you are also using that. If not, you can get a USB 3.0 to 2.0 header adapter for a few dollars or so off eBay, and use it to convert the header into a 2.0 one. You would want to make sure you get one with the metal pins on the smaller 9-pin 2.0 side, and holes on the larger 3.0 side if you did though, as adapters with the opposite arrangement also exist.
If you don't have a free USB header of any kind, but do have a PCIe 1x or larger slot available, it's also possible to add a USB PCIe card. Most of those add additional external USB ports, but some also include an additional internal header (or only internal headers in some cases). Those tend to start around $15 or so on sites like eBay, though you may be able to find some for less if you are willing to wait longer for shipment from China. Most USB 3.0 cards with an internal header will probably have a 3.0 header though, so an adapter may still be necessary. Some will also have a plug to receive power from the PSU though, so don't mistake that for a USB header.
Another option would be a USB 2.0 header splitter. Those will plug into a USB 2.0 header and split it into two or more headers, just like a USB hub, only for internal headers. They start around $8 or so. Something like that might be the most compact and straightforward option, if you don't have another USB header to spare.