Number 1 sounds weird and awkward, but if you said it to an english speaker they would understand you. Number two is a more natural sentence.

Actually, they are natural together - answering the question in the present perfect, then talking about the past: 'Have you started your new job yet?' "Yes, I have. (1) I started it last week. (2)" That's really natural. :)

Thank you so much,Jonathan and Robert!!!

The verb start suggests an action at a single point in time, but the present perfect is used for continual action that began in the past and continues now. You could use start with the present perfect when the action of starting happens more than once, I have always started my day with a healthy breakfast. So @jonathan is correct about sentence 1 being wrong. If you used the verb work, you could say, I've worked since last week.

1. Is wrong because you've got a time marker pointing to the past (i.e. "last week"). "I've started it," would be ok, "I've started it now/already," would also be ok.