'Come in' means to 'enter', where the speaker or writer's point of reference is within the place in question.
For example, if someone knocks on the door of your room, you would say 'Come in'.
It can mean 'arrive', but only if the speaker or writer is inside the place in question. For example, if you are in a station, you could say that 'The train from Oxford has just come in'. (ie entered the place where you are). Or if you are in an office, you could say 'What time does Mr Brown usually come in?'
I can't think of a situation where it would mean 'finish', though. Can you give an example?