If we're talking Ren'Py, there are 2 main ways a dev can release an update. They can release the whole thing, which will be the whole game from the start, until the end of the update, or they can release the update as a patch that needs to be applied to the previous version. In the case of a full game, just download it and play. Doesn't matter where you put it on your computer, it should be able to find and load your save files from the previous version either way. If it's a patch, just copy the game folder over the top of the existing game folder, and it'll add the new content. Again you should be able to load your saves.
Whether your saves work or not is down to the changes the dev made in the update, not how you "install" it. If they say old saves are incompatible, then there isn't really anything you can do about it without unpacking the game and looking through the whole thing to work out why they're incompatible, and then modding the game to fix it.
As for missing out on choices, using an "incompatible" save could possibly result in this, but it depends heavily on how the game is made, and this is assuming the old save works at all. Sometimes it'll just throw an error at you as soon as you load it. If you're worried about not getting the proper experience, best to start a new save if the dev says old ones won't work. No need to keep previous versions of the game though once you've got the new one, except for mod compatibility purposes.
So in the case of Becoming a Rockstar, just download 1.0 and play it, and it should find your save. If the dev says old saves are compatible, then they should work fine. If the dev says they aren't, then they won't, to put it simply.
Again, this only applies to Ren'Py games. I know nothing of how saves work in anything else.