Basically, your domain would continue to use the name servers of Company A.
You would like to create an hosting account with Company B using the same domain name. They would usually provide you with the shared IP address to which your account is assigned, but otherwise, contact their support to ask what is the shared IP address you have assigned.
Once you have received the shared IP address (188.8.131.52), you can update the host file of your system (the procedure will vary depending on your computer's operating system) to direct your domain to that shared IP address (184.108.40.206). Once this is done, loading the domain through your computer would direct you to Company B's server. Anyone else visiting your website would continue to upload Company A's server data. When you are satisfied with your new site Web, you can then update your domain to use Company B's name servers.
However, take note of any other data that may need to be transferred. For example, email data, custom DNS entries, and so on. These must be copied separately. Your new hosting provider (ie company B) should be able to help you.