You will almost always find hand washing facilities - sink, water, probably soap. What you might want to carry with you is a handkerchief. Not all public restrooms will have towels, paper or otherwise. I usually carry a mini-bottle of hand sanitizer as well, just in case. I have never not found toilet tissue but usually have a couple of Kleenex in my pocket for allergies so wouldn't get caught out anyway.

The biggest problem for older, more arthritic folks like me is the floor-level, Japanese-style toilets. Most modern facilities like department stores, larger train stations and hotels will have western style toilets. Older places, maybe not. Plan accordingly.