1. On Time

ON TIME = at a fixed/planned time established (not before, not after); e.g.

i) My train is on time, so I’ll have to leave just now.
ii) The train arrived at 5.25 – exactly on time.

2. In Time

IN TIME = not late; doing something ‘in time’ means doing it before a deadline; e.g.

i) I could not get admission because I didn’t submit my application in time.
ii) Make sure you arrive in time to see the beginning of the film.

INCORRECT: I had to rush to reach school in time.
CORRECT: I had to rush to reach school on time.

NOTE-I: If we want to convey that something happened with a comfortable margin (means well before of an event) we use ‘IN GOOD TIME’ or ‘IN PLENTY OF TIME’; e.g.

I reached the cinema hall in good time.

NOTE-II: We say ‘JUST IN TIME’ or ‘IN THE NICK OF TIME’ to emphasize that something happened immediately before the limit/deadline; e.g.

i) I could leave home a little late and arrived just in time to catch my bus.
ii) In movies, a bomb is disarmed in the nick of time, with just a few seconds left to explode.

NOTE-III: We do not use the word TIME after NOT LONG or NOT BE LONG; e.g.

INCORRECT: The dinner won’t take long time, it’ll be ready just in five minutes.
CORRECT: The dinner won’t take long, it’ll be ready just in five minutes.