Your equation is correct in a Newtonian universe, but does not consider relativistic effects.
Generally when we talk about an object's acceleration, we are talking about its acceleration in its own reference frame. By definition, an object's velocity in its own reference frame is always 0, so acceleration does not become more difficult (ignoring the fact that the surrounding matter will be impacting you at higher speeds). From the perspective of an outside observer who is not accelerating, time would appear to slow down on the accelerating object, and the outside observer would record a decreasing acceleration.
2) Even though you don't observe yourself moving at increasing speeds, you do observe the rest of the universe moving at increasing speeds. This means that small particles impacting your ship will hit it with increasing energy.
it would take a year to get to the speed of light at 1g (if my equation was right.) Then you can't speed up anymore.