(I'm assuming "..don't want to port it to Linux.." is a typo for "..from Linux" and that you want the code to run in Linux as you said in your first sentence. This means cygwin or mingw would only be used as cross compilers and aren't going to be very useful.)
This program already builds and works (or mostly works) on Linux, right? No reason to change that. Use your preferred editor (probably the one in your IDE) to edit the files and then just run the build system (probably make) in a Linux system (possibly in a VM). Export the files using a samba share (especially easy from a VM) so you can edit and automatically save remotely.
Note that you seem fine ditching every other feature of your IDE (debugger and compiler, mainly) and just using the editor part anyway.
Ah, are you not starting from any existing project and want to write this from scratch? If so, porting doesn't make any sense. You want to write cross-platform code. (Cross-platform or "portable code" being related to, but different than, the act of "porting code" from one platform to another.)
The code is then both "Windows code" and "Linux code" at the same time, and you can use any compiler on Windows that can accept the code. Usually this happens by you sticking to standard libraries and other portable libraries, or writing shims for non-portable libraries to give them a portable interface, with the compiler supporting the C++ Standard.
You can use your preferred IDE's compiler and debugger in this case, and don't need cygwin or mingw. (Unless they're used by your preferred IDE. :P)