If you have a legitimate copy, the update is small and includes:

By ensuring your Paper Mario: The Origami King installation includes the latest update data, you protect your journey from unexpected glitches, ensure smooth performance, and can focus entirely on unfolding King Olly’s sinister plot.

Here are the key "features" and fixes included in the official updates:

Regardless of which emulator you choose, ensuring you have the 1.0.1 update installed is the first step to a stable and enjoyable playthrough.