I live in Tokyo - on the surface certain lines have dated exteriors on their trains and stations, but they are maintained so much better and built with much more care to the surrounding environment that it's impossible for me to compare with NYC.
NYC's trains are constantly late, always go out of service, are deafeningly loud, incredibly slow, filthy, don't have great coverage outside of the Manhattan core and generally unreliable. Tokyo's are always on time, very clean, pretty silent, have incredible coverage of the entire metro area especially in combination with the bus networks (which also are efficient, clean, and don't hold lower-class stigma), and are super fast.
The lines that aren't underground in NYC tend to be depressing under the tracks and pretty unlivable around them, leading to tons of urban blight; the over-ground lines in Tokyo, thanks to their thoughtful design and reasonable noise level, are often full of shops, restaurants, bars, and walking paths that people actually use, positively contributing to their neighborhoods.
The main thing that feels "old" about Tokyo's trains is just that they look like they were decorated a long time ago or are kinda plain; but from a functional perspective they outclass practically any system on the planet, especially when taking into account the integration into the broader Shinkansen network. They have the best safety record of any rail system anywhere and are economically sustaining. The only major complaint I have is that it's relatively expensive fare-wise, especially considering transfers between different rail companies' lines/buses.
Also, it's worth checking out the pre-Olympics revamp of some lines - for instance Ginza Line (Shibuya station especially) and Odakyu Line (Setagaya Daita-Shimokitazawa in particular) got a lot of station and neighborhood renovations that give them that sparkly new shine.
Aside, I've also lived in Shanghai before - the rail network in China is something to watch, their high speed rail stations and urban networks are very nice as of late, although they need more express lines within city limits (Shanghai/Beijing are enormous so crossing the city by rail takes a long time).