If the domain is sitting with the same domain registrar you use then a free push of the domain into your domain account is possible, if not then the domain needs to be transferred which can take a few days.
As for moving a blog, it's simply a case of dumping the database into a .sql file (best gzipped) and using tar to archive the site files. Then once you receive the database file and site files you import the .sql file into your own database then place the archived site files into your document root for the site.
At
Fast Web Plans we can do all of this for you.