So here is the longer answer and I just added some more things, just in case.
Of course, this is just my personal view and some firsthand experience. Others may have different experiences and views on this, also some if it applies more to the Greater Tokyo area, if you are in Okinawa, the public transport aspect is almost irrelevant .....
Do a lot of research online beforehand, to get a good idea of what is out there. When interested, first check from the outside by yourself, also check if the area might suit you. Once you are seriously interested, involve a real estate agent. Of course you can involve one or more agents from the start, as they might know about not yet public properties.
Avoid buying a new apartment, if you buy one that is about 10 to 20 years old, you can save 30 to 50% percent (maybe that changed recently, depending on the area), at the same time, I would not buy anyhing built before around 1985, maybe even before 1990. I would also - just my opinion - not buy an apartment in a Tower Mansion, too many risks involved.
Decide where you are most willing to compromise. In the cities, most important for many people is close vicinity to a (good) train station, no matter how ugly, smelly, loud the area is, it is really amazing), that drives prices up 20 to 40% already. If you can live a little further out, that will save you money. The same is true for “not so good areas”; as there are almost no really bad areas in Japan, I would avoid the good = expensive areas which are not necessarily nicer, sometimes the “good" and "not so good" areas are very close together even.
Decide if you want a non-renovated, semi-renovated or fully renovated flat and how much work you are willing to do/have done before moving in. Fully renovated flats are often very cheaply done and can be depressingly dark and ugly. In the case of non-renovated flats you will be able to change the whole floorpan, a friend did this a while ago, with impressive results. (there are companies specialized in this).
Of course avoid areas that can be flooded, hit by tsunami, can suffer from liquefaction, sit on unsecured hills, can be hit by landslides - can be a bit of a challenge ….
Check if there is a chance that your view can be blocked - which is often the case if a nice old building with garden near-by is suddenly demolished to be replaced by a multi-story building.
Check the out- and the inside during different times of the day, how loud is it at nighttime, how dark during daytime etc.?
I looked also for something where I can avoid use of the aircon as much as possible by being able to use natural draft.
If you want to use a gas heater, is there a gas connection in the living room?
Do the windows have the old cheap alumunium frames? Then your room will be flooded during major rainfalls and it will be very drafty…..
How many built-in closets are there? (it seems newer apartment tend to have less).
How much storage room and work space does the kitchen have? (recently there is a trend to open kitchens, connected to the living/dining area that have very little storage space (no more overhead kitchen cabinets) and no barrier so that all the dirty dishes/ disarray can be seen from the living area, definitely only looks good when unused.
Is the access to the premises secured, (very convenient if you tend to be absent for longer periods).
Are there any shared facilities (only in expensive and newer places), e. g. a gym, a pool, rooms/a kitchen that can be rented for events, rooms that can be rented for overnight guests, a playroom, etc., is there a concierge? all this will drive up the maintenance fee quite a bit.
Regarding the discount you asked, that depends of course on the demand, ours was on the market for some time, still the discount was only a bit more than 2% (the first offer had been rejected, which seems to be normal practice) I am not good at such things and followed the agent’s suggestion, you might be able to get a much higher discount, or it might fire backwards, hard to tell, without knowing who else is interested to buy.
Hope that helps a bit.