Possible? To crazy to contemplate? if yes and no (respectively) any idea how to go about doing this?
Shouldn't be too challenging; just use Cygwin if you'd like to compile it.
I don't know if it's possible. With newer versions of Builder and some probably not-insignificant effort, I'd guess it probably is. But why on earth would you want to? You'll end up with a version of MySQL compiled in an untested environment instead of the distribution versions provided by MySQL themselves that gain lots of testing exposure. I can't think of a single advantage of doing this, but I can think of plenty of disadvantages.
If your problem is that Builder is the only development environment you have to hand, be aware that you can compile using Visual C++ Express, which MS will give you for free.
If you really want to go about doing this, you'll be wanting MySQL's build guide and CMake. Following their instructions for Visual Studio will work to get you something you can start building. You'll then need to hack around looking for appropriate compiler options and so forth.