I'm not sure if this is still a relevant issue for you or not, but if you want to simply upgrade your 10.1 site to 10.3, you can do an in-place upgrade and install 10.1 on top of 10.3 to upgrade it. If you want to move your 10.1 site to a separate machine and then upgrade to 10.3, you can disconnect the machine from the site, (under Manager > Site > Machines), and then copy the site folder, (by default C:\arcgisserver), to the new Server. You'll be redirected to the Create a site or Join a site page for the original Server site, so select Create a site and enter the same exact path as was originally set for your site. You're simply telling the installation to look at the existing content for the site. On the new machine, install 10.1, and set the site to point to the copied arcgisserver folder. Once that's done, you should have identical sites at 10.1 on separate machines. On whichever machine you want, you can then upgrade to 10.3 by running the installer.
