The option to change the time of day can be found via GAMEPLAY settings once you've finished the main story in your current file.
If you start a New Game+ playthrough, this setting cannot be changed until the story is completed in your selected file, i.e. you cannot change the time of day when the story is in progress because certain times of day are tied to the story.
There is currently no way to change the weather in Marvel's Spider-Man 2.