I have my server to auto restart at a specific time, but unfortunatley, it does not save before doing so, making players lose some progress. I looked for plugins that would save before restart, but I could not find one. Does anyone know of any?

Pretty sure the server should save when restarting, it could be that the system you are using to restart the server just kills the server process and starts it up again instead of restarting properly.

That would be correct. The Rust server automatically saves on proper shutdowns. If you are using a control panel that forcefully kills the process, that will never save.