How to transfer WinRAR key to Linux

order.htm states:

WinRAR, RAR for Unix and OS X now use the same registration key format, so you can use the same key with current WinRAR and RAR versions for all mentioned platforms.

so Yes, it is possible.

Further, it also states where to copy the key:

If you use RAR/Unix and RAR for OS X, you should copy rarreg.key to your home directory or to one of the following directories: /etc, /usr/lib, /usr/local/lib, /usr/local/etc. You may rename it to .rarreg.key or .rarregkey, if you wish, but rarreg.key is also valid.