Pay the departing sysadmin for their time, by any legal means, to provide additional information. I've had to work with companies where a core admin had just departed, and had to help hide that we then hired one such admin as part of our company with a different title in another group, partly so we could tap them legally for information about their old company's environments. We got a good engineer, they got a good contract to help out while they looked for a permanent role, and were able to factor in undocumented aspects of the old company's security practices and backup systems which they were flat-out lying about.
Find out why that admin is leaving, without their manager in the room or any witnesses. Don't take "no" or "we'll get that to you" as an answer: go behind the company's back if you have to, because if they're hiding it, it's probably _vital_ to know about.
Do a complete hardware inventory, both of material they're directly responsible for and of devices _connected_ to those. Include the names of the people responsible for services, and who need to be contacted for issues, for every single system.
Verify that the backups are complete and that they do in fact work. This is a very good time to get that backup server, or that failover switch, that has been awaiting the right time to install, and ideally perform the restorations on those.
Warn the managers that there are likely to be service interruptions, and ensure that the monitoring system works well to report them.
Do not change the default scripting language or configuration management system or source control system or account management tools until an opportunity to learn the old one is at least 80% completed.