It can be done but you would probably have a difficult time keeping the urls the same. You would probably have to have a plugin that could redirect 404 pages so you don't lose link juice. There are some benefits to doing it but it would take time.

If it was me I would either set the new site up on a new domain and prevent google from indexing it then when it was ready I would point the old domain toward the new site or create the new site locally on computer then upload it when it ready.

I know google is starting to give higher ranks to mobile ready sites and sites that support accessibility.

I actually think it would be quite straightforward. You know that word press will co-exist with static pages. And you can give your wordpress pages any url in the slug.

I will make some assumptions here that you have standard shared hosting, Cpanel, Fantastico.

1. In Cpanel, create a new subdomain. If your domain name is example.com, make a subdomain like new.example.com.
2. Do an install of Wordpress in that directory, using Fantastico or whatever.
3. View http://new.example.com to verify.
4. Look at the files and directory names in your static website.
5. Just copy everything from your static site over the wordpress site. I will assume there are no conflicting names with the wordpress files and dirs.
6. Your static pages should be working, like http://new.example.com/index.html, http://new.example.com/cats.html
7. Setup your modern Wordpress theme, menus, header.
8. One by one convert your pages. The home page might be the most work.
9. For your cats page, make a new WP page with slug /cats. Put your old html in the body. And so on.
10. When it is good enough to go live, rename your subdomains. And you can keep working on the less popular pages.

You will need 301 redirects so the /cats.html goes to /cats.
You will also need a special rule for /index.html to go to /.

Your root .htaccess file needs something like this included:

That's an outline anyway.