Hello rugved113,
We see you have a question on updating your older iPhone before switching to your new iPhone 13, and we can help out there.
With regards to updating the older device, that won't be required, although it's strongly recommend to make a new backup to iCloud and/or a computer ahead of time.
This page has the details on things to do before/when transferring data to the new device: Transfer data from your previous iOS or iPadOS device to your new iPhone, iPad, or iPod touch
Restoring a backup from a device made using an older iOS version than what the new iOS device uses will be OK. Sometimes the backup is made from an iOS version that is newer than what is on the new device, and that'd be an instance where updating a device would needed.
Take care.