Not that manicured yet, in most places, but getting there, unfortunately. Generally, lawns in public places get beaten paths in them, are walked over where a logical shortcut appears (because the urban planners never got the Japanese knack of paving the paths people are let to make themselves). They are often driven or parked over on the edges, because there's now almost half a century of parking shortage in most cities, and it's not even the lazy bums who can't walk 10m - quite often it's the delivery trucks with no other way to come close to unload, or long haulers who just park anywhere near the place they eat or sleep.
The lawns generally don't look too nice, except in parks maintained by cities or around larger buildings where there's a gardener on staff. Specially the parks near railroad or bus stations used to be littered with garbage (cigarette butts and empty packs, bottles, broken bottles, condoms, used pads etc), but that's mostly history now, they're much cleaner now than I remember them.
Amazingly, there's always some green grass, even in winter - if snow covers it. Probably because the lawns aren't American-style monoculture, with a single species of grass, but a general mix of local grasses, so at any time there's at least one kind that likes the weather. If you see a yellow lawn, it's either someone's experiment with herbicides, or a severe draught over many weeks.
People generally have lawn mowers, mostly electric (because the neighbors would complain that two-cycle fuel stinks), and quite often kludged from an old washer motor with hand-made blades, though these are a vanishing species, the regular ones coming cheap enough nowadays.
That's true for villages, too, where the traditional lawnmowers were geese. However, the bird flu scare was a pretext to introduce, around 2006, some stiff laws about growing birds (they must be caged (!)), so the peasants simply gave up and stopped growing them altogether.