Wood – How to do with this bare spot on the step

Looks like cedar, that is a good wood to use. One thing to do if you replace it, is seal the ends and backs of the boards before you put the wood on, so this is less likely to happen again

Remove the entire shower door and frame and scrub door opening tile and door/frame assembly completely clean and let dry. Re-install shower door properly and completely seal all areas where frame and tile meet with high-grade silicone caulk. Most swinging shower doors are supposed to have a vinyl flap at the bottom to prevent water spray from escaping under the door, make sure yours has one.

Cut back the floor covering to expose all damaged and soft wood. Gouge and chisel out all damaged and soft wood. You may have extensive additional damage, this answer does not address that. Use a high quality 2-part wood repair epoxy (it's kind of like auto body filler) to fill all damaged areas. Plane/sand smooth. Cover with entire new vinyl floor or make a patch somehow. This should help until the remodel. In future you need to address water issues immediately to prevent similar damage.