Hello,

Yes you should be able to run that on an Ubuntu 16 droplet.

To install MSSQL(mssql-server) follow these steps here:

• Become root:

sudo su

• Import the public repository GPG keys:

curl https://packages.microsoft.com/keys/microsoft.asc | apt-key add -

• Add the Microsoft SQL Server Ubuntu repository:

sudo add-apt-repository "$(wget -qO- https://packages.microsoft.com/config/ubuntu/16.04/mssql-server-2017.list)"

Note: you could change the 16.04 part with the exact Ubuntu version that you are suing

• Exit superuser mode.

exit

• Install SQL Server:

sudo apt-get update
sudo apt-get install -y mssql-server

• Run the following commands to install SQL Server:

sudo /opt/mssql/bin/mssql-conf setup

• After the package installation finishes, run mssql-conf setup and follow the prompts to set the SA password and choose your edition.

systemctl status mssql-server

Source: official Microsoft documentation

Regarding the .NET core install, you could follow step one form this article here:

https://www.digitalocean.com/community/tutorials/how-to-deploy-an-asp-net-core-application-with-mysql-server-using-nginx-on-ubuntu-18-04

Hope that this helps!
Regards,
Bobby