Sync Local site and Distant site

It sounds like you need to update the database to replace the guids and base URL.

I recommend WP Migrate DB Pro to quickly sync two databases. Otherwise you need to run a few SQL statements to find and replace the old domain with the new domain. In PHPMyAdmin run the following statements from the SQL window. Replace old-domain with your exact distant domain, and the new-domain with your complete local address.

UPDATE wp_options SET option_value = replace(option_value, 'http://www.old-domain.com', 'http://www.new-domain.com') WHERE option_name="home" OR option_name="siteurl";

UPDATE wp_posts SET guid = replace(guid, 'http://www.old-domain.com', 'http://www.new-domain.com');

UPDATE wp_posts SET post_content = replace(post_content, 'http://www.old-domain.com', 'http://www.new-domain.com');

Doing so will solve 90% of your link issues.