Unlike stainless steel or aluminum putters you typically will find at retail, raw carbon steel will rust.. but only if you let it.
You may prefer to keep it looking clean and uniform. Or, you might want to see it change naturally with patina over time and use. Taking care of your Goodwood putter is easier than you might expect. Below are some do's and do not's to keep your putter looking the way you want.
Keep in a dry environment
Leaving your putter in a wet headcover, damp basement, leaking car trunk, etc. will cause rust.
Dry off after use
If you play early in the morning when there is dew on the ground, play in the rain, or any other typical wet conditions, try to wipe off excess water so that it is not transferred to the inside of your headcover.
Prevent rust with oil
If you know you will be playing in damp conditions, wipe down with oil prior to your round. A good coat of oil will act as a barrier between the steel and moisture. Spraying the inside of your headcover with Putter Butter is also good practice.
If your putter is showing rust or patina that you do not want, soak in a bowl of white vinegar, and lightly scrub with a soft toothbrush or fine steel wool. This applies to raw or bead blast finishes only. Doing this on a gun blued or torched and oiled head will remove the finish.
Let it go
If you want a rusted/patina look to your putter, less frequent maintenance (as listed above) will allow your putter to evolve over time. Once it gets to a point where you are happy with its appearance, resume regular care to maintain that look.