No Cross-save for the Nintendo Switch version ya
Monster Hunter Rise is coming to PlayStation 5, PlayStation 4, Xbox One, Xbox Series X|S, and Xbox Game Pass on January 20, 2023. While the launch date is closing, Monster Hunter has tweeted about the FAQ for Monster Hunter Rise.
Monster Hunter Rise FAQ
The following image explains that PS4 and PS5 versions are bundled. Therefore, there is no need to buy/upgrade to play Monster Hunter Rise on the next-gen console. Besides, Cross-save is also available for Xbox One, Xbox Series X|S, and XGP so that there's no worry when switching between platforms.
However, the Switch version is not supported regarding the Cross-save function for Monster Hunter Rise. Last but not least, the frame rate for Monster Hunter Rise is variable depending on the performance and hardware of your setup.