An interesting use for quantum entanglement, but not one that is likely to bring about time travel.
One problem is that you are using the word "faster" in two different senses; calculation speed is not the same thing as transmission speed, despite using the same word. It would become possible to build a very big computer, with no cooling problems, possibly not all of it on the planet, but would not decrease the latency of semiconductor switches, so calculation speeds would not improve much.
But, if the word "instantaneous" has some meaning, there is a flaw in the theory of relativity. Not over surprising; theories tend to contain them, as they are created by limited human minds, and Einstein hit closer to "reality" than anybody else for quite a while. But if simultaneity can exist it is possible to define a universal frame, and relativity is based round the premise this can't be done. "Time travel" this way assumes information is limited by the speed of light; if this is not the case, the theory falls apart.
The book to read about computers that receive E-mails before they are sent is Gregory Benford's "Timescape"; I don't think the idea has been done better. Indeed, information, as massless and energy free is the thing we are most likely to be able to send through time, far more so than historians or recording cameras.