Now whenever my EC2 instance starts, it shuts down immediately, anyway to salvage this situation?
I've not tested this solution, but this is how I would go about it.
Since your instance persists data, I would guess there's a persistent volume associated with it. You can use the console GUI or the CLI tools to get this volume ID.
Then I would create a temporary new EC2 instance and attach the existing volume as a second disk. Once the second EC2 instance has booted you'll be able to
mount the existing volume, edit the crontab entries,
Finally, you can destroy the temporary EC2 instance.
And now your existing instance should be able to boot cleanly!