To answer your question directly, "is it real, or is it fake?", that is definitely an authentic Omega Speedmaster Professional watch, not some Chinese clone or something. It appears to be a reference ST 345.0022 (PIC code 3570.50), with superluminova hands/dial, so post-1997 production. Whether it is exactly what its advertised to be, or if it is all original, if it is operating correctly, when it had its last service, and whether the price is fair given its overall condition none of us can say given the lack of additional information. I don't see anything particularly off from the pics based on a cursory look. I will mention, that if this is your first Speedy, you may want to take into consideration that many of us prefer the earlier reference that used Tritium for the lume material, which can have a very attractive patina compared to the stark white superluminova, and can be had at similar prices. Just a thought.