He told me, Shall I go to the picture?

He asked me if he should go to the picture is the correct answer.
Option 1, "He asked me thathe should go to the picture," shows that the meaning of the original statement has been lost. The word 'asked' should have been followed by 'if' (since the sentence's answer in direct speech will be either 'yes' or 'no'). Furthermore, that will not be used. As a result, this choice is wrong.
In option 2, we can see that the conjunction 'if' has been inserted. However, the modal (shall) has not been demonstrated to be transformed into 'should,' making this option invalid.
Option 3 is correct: "He asked me if he should go to the picture."
Option 4, "He requested me to go to the picture," is wrong. This is due to the addition of the verb 'requested' and the absence of 'should' changing the meaning of the statement.