There is no massive fundamental/elementary "(anti)particle" (or better field) in our Universe that can move at the speed of light. The fastest known massive fundamental (anti)particles are ultra high energy astrophysical neutrinos which can move almost at speed of light but not exactly. The reason for this is that first of all they are the lightest known fundamental (anti)particles in Nature and are given high boost factors at some astrophysical sources as they're produced. So, there is no way for any massive "particle" in our Universe to travel yet faster than speed of light. The only (anti)particle that can travel at the speed of light (but not faster than that) is light itself, namely photon field. Since photon is its own anti particle, we can conclude that the only anti-matter travelling at speed of light is 'light.'
By the construction of Big Bang theory and creation of Universe from a singularity, the arrow of time has been always forwards in an expanding Universe. So, nothing in Nature can move opposite to that arrow (or backward in time as you called it.) The Feynman diagrams--which have been used to work out perturbation theories--make use of the time symmetry where as you mentioned you can replace an incoming electron with an outgoing positron and that has nothing to do with the reality of the processes being involved. They are just mathematical constructs.