Replacing a cabinet face
First of all, if anything is rotted, there's a water problem. Make sure that's fixed first.

As for the piece that's circled, if that is rotted, then I'm wondering if the water problem is bigger than we think. How's the floor in that area?

As for replacing that one piece, there's no easy way to do that as it's likely screwed/nailed and glued. You can probably cut it out and replace the piece that way, though that would only look OK if you are painting everything.