How can I get Sql Server 2016 on a 32-bit server?
How can I get Sql Server 2016 on a 32-bit server?


You can't, SQL Server 2016 only comes compiled for 64-bit.


  Is there a way to get around it with compatibility levels? 


Thankfully, no.


  I need to upgrade it to SQL Server 2016 in order to log ship to a new server, so we can put this thing out of its misery. 


You'll need to have someone install an OS supported by SQL Server 2016, then install SQL server 2016.


  Question: how can I migrate the data to a SQL 2016 server with &lt;1 day of downtime? 


Install a supported OS for SQL Server 2016 on the old 2008 server. Install SQL Server 2016, then setup log shipping as you normally would per your above quote.