Zen Cart is largely programmed in PHP which needs to run on a server. So, the quick answer is that it is easiest to run it on your web hosting.
What you need then is a way of downloading and uploading files from the server. Basically, you download a file and then edit it and then upload it again.
The software that lets you upload / download the files is a ftp program. There are a few around and everybody has their favorite.
You can edit the files with any text editor. Notepad on windows for instance. But there are programs that make the editing easier. Again everyone has their own favorites.
Then there are some programs that combine the editing and the uploading / downloading capabilities.
I am not a windows user so I won't make too many suggestions unless you happen to use linux. But there are perfectly serviceable free options for all this stuff.
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There is a feature in admin that allows you to put the site completely out of action - ie under maintenance - which is in admin>>configuration>> maintenance.
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You can work on a local copy on your own computer but you need to install a local server ( which acts like a server on your own computer ) such as Xampp. This is not too difficult but may not be needed for what you want.
Hope this helps