Is there a proper way to apply the updates? I get an error message on reload after installing the update.

Error while compiling: ZLevelsRemastered.cs(15,0): error CS1028: Unexpected processor directive (no #region for this #endregion)

Is there anything additional that needs to be done?

You will have to specify what type of server do you have , is it a hosted server and do you have full ftp access ? , if it is all you do (if you have a dashboard)  is go to your plugins folder and delete the .cs file then upload the update .cs file from your downloads folder and you should get a confirmation that the file has been loaded successfully.
If you dont have ftp access you will have to ask the server host to do the above for you.
If you have a home server you will have to wait for someone who runs one how to proceed depending on your os

I have FTP access - I did what you suggested and that's how I got the error. I'll be sure to include those details next time, didn't think any of that would matter for updating a plugin.

ItsNatoriousB

I have FTP access - I did what you suggested and that's how I got the error. I'll be sure to include those details next time, didn't think any of that would matter for updating a plugin.

I can not see how it would not compile if you followed those directions, make sure you deleted the old .cs file in your downloads folder and that the new one does not have (1) at the end of the file. other than that i would repeat the process by deleting the file in the downloads folder and redownload the update, then delete the .cs file in your plugins folder and upload the new update.