They have those doors in many places in Asia also, but most of the trains people are jumping in front of in Japan are above ground and the stations are literally just platforms with a roof.

Platform doors are certainly possible above ground too.

Usually in Tokyo, they use chest-height barriers/doors for above-ground lines, e.g. currently the Meguro-line has these, and in the future, the Yamanote-line will. This doesn't make suicides impossible, but makes them more difficult to a degree which will certainly discourage many people (you can't just fling yourself off the platform at the last moment). [Some subways, e.g. the Fukutoshin-line, use these as well.]

A more problematic issue is stations which handle many disparate types of trains on the same platforms, which makes it much more difficult to achieve the consistent door positioning you need for platform doors.

As a guy sitting in a soba joint in Tokyo right now...I cannot imagine how much it would cost to put those doors on all the JR and Metro stations. It's certainly in the realm of prohibitive.

They do end up saving some money too, though, not only by reducing suicides and accidents, but because platform doors seem to be considered a prerequisite for "one-man operation" (ワンマン運転) on most lines, and that can help reduce staffing costs.

Most Tokyo stations also have high enough ridership that such capital improvements are a lot easier to swallow than they would be for "concrete slab in the countryside" stations.

Luckily, it's the sort of thing they can do incrementally in the places where it makes the most sense; no need to do every line and every station immediately. I imagine they'll probably continue to do it slowly, just as they're slowly adding elevators and other accessibility improvements to even minor stations.

Some metro and train stations have those in Japan too, though they are still somewhat rare sight.