You should probably invest in a wireless router. That doesn't necessarily mean the desktop PC has to use wireless. It could still be wired to the router if you prefer. But that would get the phone connected to your network over wireless, at which point the desktop and phone could communicate (i.e., sync). Of course, if you want to consider making your desktop wireless too, you'd need to install a USB or PCI wireless adapter. Up to you.
Virtually every wireless router includes 4-5 ethernet (LAN) ports so you can plug in various wired-only devices, such as your desktop. Once all your devices are using the same router, whether they are connected via wire or wireless, they are "networked" and can communicate with each other, including syncing. IOW, the phone and desktop don’t need to be specifically connected over wireless to sync. They just need to be part of the same network established by the router, regardless whether it’s wire or wireless. Adding a wireless adapter to the desktop is not a necessity, it’s optional, it’s just a convenience. But if you’d like to have a wireless desktop too, yes, buy a wireless adapter. But again, a wired connection for the desktop is all that’s absolutely necessary for sync purposes.