I can't think of an idiomatic expression specifically evocative of a repeated activity leading to "immunity", but for the more general case (which may be a one-off, repeated, and/or prolonged activity), it's...
toughen someone (up) - to make someone stronger and more able to deal with difficult situations
Here, for example, are as few thousand instances in Google Books of "toughened him up". If you look at a few, you'll see that although it's often used in contexts where the "toughening" refers to becoming more physically "robust" (stronger, better-toned muscles, etc.), it's also often used of "emotional" strength.
EDIT I deleted the above answer when OP clarified that he was looking for an English expression for exactly the meaning being "toughened up" by repeated exposure to cold water. Obviously my answer doesn't get very close to that - as I said, it doesn't even imply repeated, let alone cold water.
I've now undeleted it because although there may not be an exact answer, this one is at least "relevant".
But I will just say that I think it's unlikely there will be a short English term evocative of cold water and "positive" effects such as acquiring immunity/becoming stronger. The reason I think it's unlikely is because any such usage would clash with a well-established idiom...
pour cold water on - to discourage doing something; to reduce enthusiasm for something. (Alludes to cooling passion with cold water.)