Can I have two independent flights?

For example: I have bought two independent tickets. One from Madrid-Shanghai and another one Shanghai-Melbourne. Can I apply for the 144 hour free transit if the second flight is within 2 or 3 days?
I know my question might sound stupid but I would like to make sure I can do it if the flights were purchased separately.
Also, I am not very sure how to apply for it despite having read the article. Should i do it the same day before departure when check in my luggage? I am afraid I might be stop from boarding.

Thank you :)