“too big to carry on the plane”
Too … to … means so much of a quality that an action is impossible. Too big to carry means too large to carry. Here carry on refers to taking luggage into an aircraft cabin; compare carry-on bag. Tina chooses to mail the item because it is too large to take aboard.
“Standard or express shipping?” / “We don't have that option.”
Shipping means sending or delivering goods. Standard means regular and express means fast; Standard or express shipping? is a typical post-office question. An option is a choice, and We don't have that option. is a polite way to say a service is not available. Here the clerk answers Tina's claim that her planet sends things by light as if it were a serious question about Earth's delivery services.
“I can get there before the box!”
Get there means to arrive there; here, in Japan. Before the box means before the box arrives. The two of them are flying to Japan the next day, so Tina is amazed that she will arrive before a box that takes about a week.