Do you have FTP access to your site or any other way to replace files directly?
Download a fresh copy of the theme you are using from wordpress.org/theme
Replace your current theme files with the new set using the steps that apply to your situation from Using Themes: Adding New Themes.
This will overwrite any changes that you’ve made to the theme and you can just overwrite the functions.php file if you want.
You can build a site to replace your existing one and use your current domain to access it.
You can sniff around to figure out how it is setup, but why not email the person who built your site and holds the domain? If your site is on WordPress.com, they will be able to supply your with the account credentials. If it is hosted by them, they can transfer your domain to your new host when you are ready for your site to go live.
