You don’t need to take the systems offline. I do such a procedure in the normal daily business.
Before upgrade test your modifications if they work in a small test system.
If all is ok - start with one of the systems - only stop site - upgrade site - start site - normally 2-3 minutes per site. Only you need to ensure that no one does any changes on the master before you finish all updates on the sites.
After all sites run the same version check if it possible to push config from master and then you can repeat the procedure with the next version step.