I've termed this type of development "Maintenance Driven Development" (http://taylodl.wordpress.com/2012/07/21/maintenance-driven-d...)

Yes, it's a good idea to write unit tests for the existing code base, though be careful with setting unrealistic expectations with regards to test coverage. The goal is to better understand the existing code base and have tests around the most critical parts of functionality to provide evidence they haven't been negatively impacted by the code changes.