In our overview of Bender's Big Score we concluded that the time travel story might begin at the end, because it is in the last minutes of the movie that Bender leaves to place the tattoo on Fry; time travel is initiated in the film when that tattoo is discovered to contain the time travel code. No one can travel to the past until the code is discovered, and the code cannot be discovered until Bender brings it to the past. In this form of grandfather paradox, a type of predestination paradox, we have information being passed from the future to the past, in essence knowledge known in the future becoming the basis for its own presence in the past, and thus for its own existence in the future. That is, Bender can take the code and place it on Fry because he found it on Fry initially. As elsewhere discussed, what matters is that the knowledge is available in the future, and thus can be taken to the past. We saw similar events in connection with Mr. Scott's transwarp teleportation formula in the recent film Star Trek, and in connection with transparent aluminum in Star Trek IV: The Voyage Home.
The problem in the Futurama version is that the original source of the knowledge has been lost, and there are few if any clues to enable us to reconstruct a history of the universe which would have led to the discovery of the code and its placement on Fry in the past.
It is almost certain that in the original history it was Bender who left from the future to carry the code; that tells us that the code came into the possession of the Planet Express Delivery Company. The question is, how did that happen? The answers might be
Time travel in the film creates several impossibilities; it seems best to assume that there was no time travel in the original history up to the point at which Bender takes the code back to Fry, or that point could not have been reached. This means the Scammers never had it, so whoever found it gave it to the Planet Express Delivery Company. Since quite independent of time travel, that company was the point of initial attack, it may have been thought that they could use time travel to prevent themselves from falling prey to the scams which led ultimately to the taking of Earth. The decision to deliver it to themselves via Fry's anatomy has merit in that they might hope their past selves would identify the use of the code before the Scammers attacked, and could use it against the invaders.
Thus the most plausible reconstruction of the original timeline has the scammers taking Earth, ultimately shipping humanity to Neptune; humanity regaining their planet by invasion; then the time code being discovered and given to Bender, who departs to attach it to Fry, creating the altered history in which Fry has a tattoo.
This, then, sets up the time travel that we see in the film, which we will begin to consider next time.