In Johnny Tremain, when was Johnny supposed to take his cup to Merchant Lyte?
The answer to this can be found towards the end of Chapter I. It can be found when Johnny is telling Cilla something about his past. There, he tells her what his mother told him about the cup. Basically, her instructions to Johnny are that he is to take the cup to Merchant Lyte when he (Johnny) is in a really bad situation. When he really needs help, that is the time to take the cup to Lyte.
What Johnny's mother told him was
...if there is not one thing left for you and you have no trade and no health, and God Himself has turned away His face from you, then go to Merchant Lyte...
From this, we can see that Johnny is supposed to bring the up to Lyte when everything is going really badly for him.