The Unix epoch problem is completely unrelated to a program being 32-bit or not. The architecture affects the maximum addressable memory space, not the size of individual types. You could easily define and use a 128-bit type in a 16-bit environment, for example.

The epoch problem is simply due to a bad design call a long time ago - one that proved foundational and incredibly difficult to change once it'd become an entrenched standard. They could have made timestamps 64-bit at the time, and probably would have if they'd known their work would survive the several decades it'd take for that decision to pay off.