Well, we would at the very least need antimatter.

Antimatter is "promising" due to its mass to energy ratio...however, it is currently the single most expensive material by mass known to humanity, so fat chance of using just AM to power anything....

More reasonable propositions use AM to start a fusion process...still expensive, but the amount of AM needed is in the micro- or milligram range...I'd say it's also fair to assume AM production costs would go down if there existed a demand for AM beyond theoretical science...

The problem here with AM is that we have absolutely no idea if it is "good enough"...we have no idea if FTL is actually possible, how it would work, and what the energy requirements would be...so, even if AM is the best option *now*, it may still not be good enough to power FTL propulsion...

If the spaceship accelerates at a constant 1g, he will after a little less than a year (mathematically) reach almost the speed of light

This is a very misleading statement...at about 86.6% the speed of light, your relativistic momentum would be twice the classical value, meaning you would need twice the force to maintain your original acceleration...at this point, your kinetic energy would be exactly equal to your rest mass energy...

Some might even argue that constant "acceleration" inherently implies a constant force when talking about relativistic mechanics...constant *force* rather than constant *acceleration* is a better model since you could never design a device that would maintain constant acceleration for all speeds (you would, in fact, need infinite energy)...

If he returns to Earth he will land thousands of years into its future.

For a 40 year journey, one would need to be traveling at about 97.98% the speed of light relative to earth in order for 1000 years to pass here...this is completely neglecting your acceleration phase, so in reality, you would need to obtain a max speed *much* more energetic....