Like WPyogi, I think the easiest way is to develop a test site on your server, not local.
Especially because most shared hosting providers provide a one click install.
The other thing I was thinking about in your case is this:
You would still need to move your site even if you develop it on a server in a subfolder.
Moving a site is not trivial as mentioned before so it may take some time to do it, and the site would be down during that time.
If your clients site is simple (not too many pages, etc) you may want to consider this:
Copy all pages of the old site to local computer. There is freeware that will copy a whole site to your local computer and keep the links, linked to local files. So you can still navigate from page to page on your old local site.
Install WP on the server in the root of the domain.
Copy and paste the contents of the old local pages to the WP live site.....
This way there is no moving involved...
It may take a similar amount of down time to 1) Move vs. 2) Copying content of pages across. Even if you do move the WP site, you will have to copy content across.
And maybe before doing this, do install WP in a subfolder to learn how to use it, make a demo, use Lorum Ipsum pages etc, to show to your client. But use it just for testing/learning. Dont move it.