Tokyo, Kyoto and Osaka subway

Hello,
I couldn't find it anywhere, does the trains on the subway in Tokyo, Kyoto and Osaka run 24/7? What are the times of the trains?

Another question, we will be staying in Tokyo for 4 days and another 3 days (apart from each other), Kyoto for 3 days and Osaka for 3 days. Should we buy a pass or just pay for the trains on site each time?

Thanks!

does the trains on the subway in Tokyo, Kyoto and Osaka run 24/7?

No, they don't run during the night (roughly between midnight and 5am).

What are the times of the trains?

You can look up connections here:
http://www.hyperdia.com/

Another question, we will be staying in Tokyo for 4 days and another 3 days (apart from each other), Kyoto for 3 days and Osaka for 3 days. Should we buy a pass or just pay for the trains on site each time?

A Japan Rail Pass won't pay off. Day passes for Tokyo and Kyoto are generally not very well priced. Osaka has some good day passes, though. Generally, I recommend to use an IC card (such as Suica) for urban travel.

You can find out more on each city's transportation page:

Tokyo: http://www.japan-guide.com/e/e2017.html
Kyoto: http://www.japan-guide.com/e/e2363.html
Osaka: http://www.japan-guide.com/e/e4006.html

Your only long-distance trip (Tokyo to Kyoto) can be done at various budgets as explained here:
http://www.japan-guide.com/e/e2363.html

Thank you very much!
Actually - I will be staying in many cities, so I have already bought the 14-day JR Pass (calculated it and it was really worthwhile).
My question was about those three cities, thanks for the answer.
If I understand correctly, the IC card doesn't offer any discount, but only convenience at the station itself before boarding the trains/buses, am I right?

The JR Rail Pass will not work in the subways. Those are private railways. Tokyo METRO has 1-day tickets, but you need to ride a lot to make it pay off.
PASMO and SUICA offer no discount but they are convenient.