Fortunately for you, I have a lot of experience living off others people's kindness. Here are some handy tips I've picked up over time.
You want to get this one out of the way right off the bat. Ideally, your host will think of that beforehand, but if not, you'd better ask before you shower, because afterwards you'll be naked and wet and not in any easy position to ask, especially if they're somewhere out of earshot. You have the option of just grabbing a hanging one at random if you like, depending on how close you are with the person in question and whether or not any old people live there too.
If you do forget and are left without recourse, you can use an item of clothing as a makeshift towel, especially if it's not something immediately necessary to your wardrobe, like if you've layered a couple of shirts or something. If it's winter, definitely use a shirt because you can keep it under your jacket and it won't freeze. If it's summer and you're in a dry climate, you can pretty much just put your clothes back on normally if you really want and they'll dry soon enough, but if it's humid, don't even try – you'll be sopping all day. Actually, you will be anyway, but this way it'll be even worse.
Again, preparation – remember to figure out how they work before you strip. That way, if you're absolutely baffled, at least you don't have to get dressed again before you can go ask for a demonstration. Once you've got it all worked out, you'll be ready to get naked, crank a knob until a stream of hot fluid bursts over your face and cascades down your chest, and exhale in ecstasy.
Some Japanese baths have an electronic control panel for the bath itself. You maybe shouldn't touch it. And actually it's probably set to the preferences of the owner(s), so you shouldn't touch it anyway.
As I'm sure you know, Japanese families all share a single dispensary of bathwater amongst them, which isn't emptied until everybody is done. Some people find Japanese bathing to be one of the best experiences available to humanity, but I've always been a little iffy about it, not because I have to bathe in other people's filth, but because I don't want to make them bathe in mine. You're not supposed to go in until you're spotlessly clean, and I just don't trust myself to be able to do that. Worse, as a guest you may be afforded the respect of bathing first.
Luckily, there's an easy fix: Just say that you would prefer a shower. Basically, you're just skipping the second half. You'll be clean, so it's not like you're being rude, and you can even invent a cultural explanation if you want. I've never had anybody insist I actually bathe, because that would be crazy. How would they check, anyway?
If you do decide to take the plunge, so to speak, obviously just be very thorough. Wash everything twice. Wash all the places you usually don't bother with (you have some, don't lie to me). When you're done the bath should be a basin of crystal clear water and nothing else. In practice even the Japanese sometimes accidentally shed detritus, but if you do, you just know it'll be because you're a foreigner and not because you're a human being, so scan carefully for any stray dirt or hair and scoop it out with your hand. There's a grate in the floor you can drop it down.
You can use this to pour water over your head, or as a little stool. I like to just sit on it and douse myself with the jet.
Shampoo and soap
One abiding principle: Honestly, they're letting you use their shower. You really think they're gonna get offended if you swipe some of their shampoo?
On the other hand, if you're having trouble with the shampoo, you don't have to wash your hair, you know. And also try to be at least a little careful that you're using your friend's (or whoever's) stuff if possible, rather than their roommate's or something. That's just called respect.
However, the preceding rule can be safely ignored if there is both bar soap and liquid soap. In that case use the liquid no matter whose it is, because which would you rather be rubbing all over your body? Liquid is better for everybody. If there's only bar though, it's not a big deal, it's not going to hurt you, because, you know, it's soap. It does the opposite of that. But! If you're still not comfortable, check to see if there's a liquid hand soap you can grab off the sink. Works fine. I only ever used hand soap during my last study abroad. Cologne once said “I picked up some more hand soap for you to shower with.”
In a pinch, shampoo or conditioner can also be used as soap – it's not as effective, but it's all cleaning agent. Just make sure to wash it all off or it can dry out your skin and leave a painful rash.
These are just a few simple shower hacks to help you with your stranger showering experience. Got a tip of you own? Let us know in the comments!