Do I need to pass Spamton NEO on a snowgrave, or is it enough to get by with a secret boss (It's the same boss)? Also in the rules it is indicated that the passage should be in one save file.

For the Snowgrave category, you must complete the game using the Snowgrave route, which includes beating Spamton NEO in the same route.

I meant do you need to go through snowgrave for the category "All Bosses"?

No