The amount of upkeep that goes into a Japanese apartment astonishes me sometimes. Coming from a country where "Clean the tub" usually means scrub out the insides and perhaps snake the drain if you really need to, I was utterly unprepared my first time trying to deep-clean the shower and bath area. If you have a tub like mine, you'll notice a great many procedures need to be done before you can call the place "clean".
Most foreigners who've at least watched the 80s flick Mister Baseball know that you aren't supposed to soap up in the tub. This cuts down on soap scum in the tub itself, but that doesn't mean the process is easy otherwise, partially because we don't stop cleaning at what your body touches, but more on that later. Japanese bathtubs tend to have greater depth than their counterparts in the west, which means even just cleaning the parts of the tub deemed normal by international standards may take more effort than it would elsewhere. If your shower room is also where your clothes dry, you might have to wipe up or pick out lint from the plug area as well.
Also, the faucet part in tubs like mine can be cleaned separately either with a solvent dissolved in bathwater and/or flushed with one of these units. I have yet to use this, but the picture really says it all.
To clean the rest, you'll need gloves, paper towels, cleaning spray, and a plastic back to throw the scum-covered towels in. I also recommend a strong stomach. In addition, I sometimes use fizzing de-clogging tablets which are included in the picture above.
First, attack the drain.
In general practice, the actions here repeat. If the part can easily be taken out or off, take it out or off. If it is gross, rinse it off before you put it back. If it's really gross, wipe it down or out with a paper towel first. Really super gross? Spray it with a cleaner, let it sit a few minutes, then rinse it and wipe it until it is clean. This goes for basically everything in the bathtub-drain area.
I do recommend taking your time so that you don't forget where things go. Also, please use your apartment's ventilation system to assure fumes from any cleaning agents don't cause any negative reactions for yourself or your housemates.
Then we come to a bigger problem-- taking off the facing of the tub. On older models, this won't be an option as the tub itself will be welded together completely, but that means there is less for you to clean. If you have a fancier tub, especially one that fills itself at the press of a button, there's a bit more work to do. The great thing is that many of these tubs have picture based label guides to show you what to take out and put back to get into the under-tub. First, take out the under-bits that wedge the facing into place. On my model, there is one of these on each side. Take each one out. This might mean pushing against the facing to take weight of the wedges so that you can retrieve them.
Once access to the under-tub is granted, spray and wipe every surface you can reach, cleaning up whatever you can. Don't be afraid to let your cleaner soak a problem area for a minute if it is too difficult to deal with immediately.
I do not recommend messing with any of the tubes under the tub. All the pieces I remove and clean are simple plastic, occasionally with rubber rings in one place or another. These are generally safe to remove, clean, and replace.
When everything is as clean as you can make it, put the facing back on the tub and wedge the under-bits back in place, following the pictures on the guide if provided. If there is no guide, look for other clues, like this kanji.
Then start putting everything back in place, slowly considering the pieces of the plumbing puzzle if necessary. If the facing of your tub is like the facing on mine, you may have to hoist the facing up to wedge the two under-bits in place. Do your best to wedge them under well. Don't worry if this part is tricky for you. I am still very bad at it, too.
The hardest part for me is under the drain, in a little tank of standing water that I really so not understand the need for in this country of perpetual humidity, but I'm also not a plumbing expert. I do find this area next to impossible to clean sufficiently due to lack of access. How do you clean something you cannot take it apart and wash/rinse the individual pieces of?
Instead I swish a gloved hand around inside, pulling out clumps of gunk and throwing them in my garbage bag before diluting the remaining gunk with clean hot water from the shower head. Usually, I add a drain unclogging tablet thing, which fizzes cleanliness into the effected area.
However you get it done, good luck in cleaning your Japanese bathtub!