No, sweet iron does rust. I have one just like that and the mouth is rusted on it as well. My horses like it.
Then only thing about rust on your bit is that you need to check it periodically to ensure that there are no sharp spots forming. If there is, just take a bit of steel wool and scrub and it will usually smooth out the point.
Actually, because I went through a stage where I couldn't find any more, I've kept my eye out and have found several bits that are shaped basically the same but with slight differences. One with a solid copper mouth and others with slightly different shaped mouths. This style seems to be gaining popularity again....thankfully .