His secretary said he would be too busy his first week back.

当一个介词后面的名词，有其他修饰语而不再单调时，可以省略介词，目的是为了简练。例如：

He arrived at ten o'clock.

He arrived (at) about ten o'clock.

网友的句子：

His secretary said he would be too busy in his first week.

His secretary said he would be too busy (in) his first week back.

个人看法：不省略是为了避免单调，省略是为了简练。