If I buy a cartridge for a specific game will I still have to download the game over the internet (like desktop console games) or just the updates? And if it's just the updates, can I download it over the internet and the game will be able to work?
If you buy a cartridge, then the Game is on it, the DLC or the updates were downloaded over the web, via Ninty Servers.
It depends on the game, the majority of games have the game in its entirety on the cartridge, meaning you only have to download updates which you do for free over the internet. Some reissues of games I believe come with the update already on the cart as well.
Some games however require updates to play fully or partially because the publisher didn't want to pay for the extra cart size or the game was rushed out for release (i.e. can only play Spyro 1, in Spyro Reignited Trilogy, Pokemon BDSP is missing its postgame and music without the update).
When you download an update you don't download the base game again (however because of the file structure of some games the update may basically contain the contents of the base game and the update may be even larger then the base game itself).
I hope my fishing games are not like this. xDSome games are so large, only half of them fit on the cart and you need to download the other half just to finish it. (Looking at YOU, Wolfenstein II.)