Ubuntu – Upgraded to Ubuntu 16.04LTS – but no new Software Center

Yes, it is but it takes a bit of doing.

old software center

Download the previous version, Un-install the latest version using synaptic package manager it is called "software-center" It will want you to uninstall a ubuntu-desktop or xubuntu-desktop, let it uninstall this package then re-install it after it is done un-installing the software center.

next close synaptic package manager and open the download of the old software center with GDebi Package Installer, If you don't have it install it first. The old version will install only if the newer version is uninstalled first.

after installing the old version open synaptic package manager locate software-center, highlight it and on the file bar click -Package -Lock Version, this will prevent it from being updated all the time.

You don't have any software sources configured, which means something went wrong with your installation. As there may be more problems I'd install the system again. Make sure you use an official up-to-date Xubuntu image.

If you want to try to repair your system you can use for example https://repogen.simplylinux.ch/ to create a new sources.list

After saving the new /etc/apt/sources.list run

sudo apt-get update
sudo apt-get dist-upgrade

apt-get should offer you to upgrade some packages. Do that, and then run

sudo apt-get install xubuntu-desktop

to get all the software installed for Xubuntu by default.