Went through this with a tiny crane my grandmother folded before I was born, so I know how nerve-wracking it is to handle something that fragile. First thing: stop trying to clean the stains. Water, wipes, erasers, all of it will do more harm than good on old untreated paper, especally something that small. Those blue and brown marks are probly some kind of staining or foxing, and honestly once they're in the paper they're staying. What matters now is keeping it from getting worse. If you have to move it, slide a thin piece of clean cardstock under it and carry it that way. Wash and dry your hands first, or better yet handle it with clean tweezers very gently at the base. Don't press anything flat, don't unfold anything, just leave the folds alone.
Long term storage is the easy part. Get an acid-free box or a small archival envelope, tuck it in somewhere dark inside your living space where the temperature stays steady, like a closet shelf, never an attic, garage, or basement. A silica gel packet nearby helps with damp air. Check on it every few months with your eyes only. Mine has looked the same for six years now doing exactly this.