If you want to save at Mt. LavaLava, you must find the hotkeys for your emulator and create a save state. In addition, saving Paper Mario with the regular save blocks works, too.
If you are having issues with saving and updating your ROM, I suggest checking the file extension of your game. If you have Project64 Version 1.6, you can save in .z64 and .v64 files, but not .n64 files.
In the case that you still cannot save, I suggest trying other methods like installing or reinstalling Project64 onto your computer. During the installation of Project64, a folder will be created in the "Program Files" menu of your C Drive (if you have a Windows operating system). Within the Project64 folder, compressed .zip folders will contain your saves for each game, so you can load to whichever point you need to resume with the ROM.
He's talking about a save state in that point of the game. I don't know of any, I had to do my whole Paper Mario game without savestates. There may be some on gaming websites but I don't know if they'd support the use of emulators and roms.