Problem with iTunes Store and updating WhatsApp

So the problem is that I have an iPhone5 from my cousin, and yesterday I got a messages in WhatsApp from friends but I couldn't open WhatsApp. It said that I need to update WhatsApp. I went to the App Store and it required the password of iTunes Store but old email and password from my cousin, and I don't know the password.

Does someone have a solution for this? I also don't want to delete my WhatsApp because I cannot backup my chat history and also will not see the new messages that come in my WhatsApp.

It sounds very much like the phone wasn't wiped & set up as your own new device.
This means that you will need the old login data for everything you need to do on the phone - this is not good.

You will need to get your cousin to make sure the phone is removed from his iCloud account, then you need to wipe it & start afresh.
You will lose all existing data, but currently, nothing on the phone belongs to you.

See What to do before selling or giving away your iPhone, iPad, or iPod touch