Can I auto...?

Can I auto update a MineOS mc server’s modpack from a ftb private modpack?
Even if it’s only by downloading server zip download and extracting it.

Well, not really. You can definitely add new profiles which can replace the existing jars, but “blindly” having the web-ui update your files on a schedule without user interaction seems a very dangerous decision.

Not so much that the new server packs won’t work–but assigning a new profile doesn’t remove the old profile, so there might be a point where plugins that have been removed from newer packs still exist in the server directory, and could cause issues.

I think updating modpacks is something that must be done with deliberate care.