Here's how it fits your description:
* Bruce Willis eats a spider: He does, in a scene where he's being interrogated and forced to eat a spider by a guard.
* Time travel: The main plot of the film involves Willis's character, Cole, being sent back in time to prevent a deadly virus from being released.
* Hook up with Brad Pitt: While they don't "hook up" in the traditional sense, they do have a tense and complicated relationship. Brad Pitt plays Jeffrey Goines, a violent and unstable man who Cole believes may be connected to the virus outbreak.
* Watch himself get shot: Cole sees his own future self get shot in a flashback sequence.
Let me know if you'd like to know more about the movie!