Can I update from CM 12.1 to CM 13 using OTA, or do I need to follow instructions?
The OTA-update will only update CM itself. You will still have to flash Google Apps and vendor.img manually. If you had Google Apps installed and only do an OTA-update, your Google Apps will most likely be incompatible with your system, causing your device to get stuck in a boot loop.

I don't have a Nexus 9, so I don't know if the old vendor.img is compatible with CM13, but I would recommend to update it as well, just to be sure.