How would you rate Japanese cities in terms of barrier-free facilities at train stations, on buses, in restaurants, stores and other public areas?

I had to check what barrier-free facilities means.

https://www.nippon.com/en/features/jg00087/barrier-free-design-in-japan.html

Basically every ramp, escalator, elevators and what else meant to help the elderly and people with disabilities.

I think Japan as a whole is doing pretty good on this point.

Things have been changing drastically in this area, and large paralympic sports events (1998 winter games, and the coming 2020 games) are a big influencer, just as the 2002 joint-hosted FIFA world cup brought about a lot more multilingual signage.

Sidewalks are still a nightmare for people with mobility issues. Bicycles parked willy-nilly. Buildings and public transport have been retrofitting and all new buildings in the last 20 years are designed to be barrier-free.

There is still room for improvement. Silly little things like elevator buttons that require blood flow to the fingers to work. (My boss with a prosthetic hand struggled with those.) And my office now, which is a new building, has round door knobs on the heavy fire doors. Round knobs are a challenge for anyone with a hand injury, or debilitating condition, such as arthritis.

I would really like to see fire inspectors do more patrolling of fire escapes in small buildings in entertainment districts. I hate going to 5th floor izakayas and then seeing that the fire escape is full of supplies and utterly impassible.

Very mixed. Newer buildings are generally good, but its pot luck whether an older building has been renovated or not. When I lived in Kanto, all train and bus stations seemed to have barrier-free access, although in some of the older places you might have to search a bit to find the ramp or elevator. Now I live out in the sticks and our local station only has a footbridge with stairs to access the far platform. Not even a gate on the other side of the station or one of those little footpaths across the track. Basically if you are infirm or in a wheelchair, you need someone to carry you across the bridge. Not good enough.

I used to live in a barrier-free apartment. Even as an able-bodied person, it was quite nice having wide doorways and hand rails. The genkan area was even slightly sloped so people could still 上がる.

Recently, I had the experience of taking an elderly wheelchair bound person from Shikoku to Tokyo by train. (There were a couple of other family members helping too.) I was dreading it, but it went very smoothly. The train company was informed in advance, and there were staff waiting at all points to attach ramps to the doors and help with getting on and off. At Okayama, a staff member helped with the transfer across the station. And the shinkansen toilet for wheelchair bound passengers was easier to use than the person's toilet at home. I was impressed.

Define barrier free please.

Depends on the station. Some are great, others are absolute nightmares...But I think the city is trying to make improvements before the olympics.