The question is should there be a separate product backlog item (PBI)/ User story for test automation in each sprint.
The answer to the above question is YES. There should be a separate PBI/ User story for test automation in each sprint.
In the first sprint when thinks are still evolving you may not want to have a PBI for test automation. But from sprint two you should have a separate PBI. I feel that there should be two PBI for test automation; one for the test automation of the current sprint items and one for the updates that you may have to do the old test scripts.
The time or the story points depends on the sprint velocity. Once the sprint is completed and the build is in QA; the automation scripts should be ready.
Prioritising the work (Stack ranking the work) is another very important task; first finish automating the current sprint items and then allocate the time to updates the old test scripts.