I just reinstalled Duke3d for the first time in awhile, and I'm disappointed to see it doesn't start. The icon appears in the dock briefly then disappears. I explored by right clicking on the package and exploring the contents, and under Resources/game dir I see that "Duke Nukem 3d.app" says that it is out of date and needs to be updated, which is Catalina codespeak meaning it is not 64bit and can't be run on the OS.

Unfortunate...

Are there any plans to update this? If not, how can I update it myself? I'm assuming DOSbox has been updated to support Catalina, after all.

I'd suggest using EDuke32 instead. You can download a reasonably recent build here https://dukeworld.com/eduke32/mac/