No - Replay does not automatically switch you over or upgrade your plan once the value of the monthly Unlimited plan has been hit. This will need to be done by the customer.
If you are doing a one-time copy, or only intend to use Replay intermittently, the Pay As You Go option would be the most ideal selection for you.
However, if you are looking to make multiple copies, maintain a continuous, automatic copy between stores, and/or copy your Product metadata, then we highly recommend Replay's monthly subscription, as it is more cost-effective.
If on the PAYG plan, we recommend keeping an eye on the Summary box presented to you during any copy task you perform, as it'll display the exact amount the copy task will charge you (before you start the task / get charged).
Should this amount hit or exceed the monthly cost for the Unlimited plan, you can simply switch your Replay plan over to the Unlimited plan at that point. There is no penalty incurred for switching your Replay subscription.