My recommendation is that you:
- Backup your live DNN 4.4.1 database and web directory and install them on your local computer. You'll need to edit the portalalias table on the local machine to make it work, and you'll have to edit the connection strings in web.config.
- Upgrade the local site to 4.5.5.
- Backup the local database and web directory and do the inverse of the first step. That should give you an upgraded live site.
- Don't forget to keep copies of all of the backups so that you can restore the various sites if something goes wrong.
You can have many installations of DNN on your local computer. Each one would be set up as a virtual directory with IIS. You can also have other, non-DotNetNuke, websites.