The times should be equal. However, if we were hypothetically talking about someone on earth observing the rocket, or vice versa, time would appear to run slower in the observed frame of reference
Your answer is correct but it seems you're over complicating things a bitThis is a lot more complicated that it looks... a rocket travelling at 0.75 c will take 9 / 0.75 = 12 years to travel the distance. But let's pretend that someone on the "distant planet" shines a green light towards Earth when the rocket arrives - then that signal would take 9 years to get from there to here. So the observer on Earth would wait 21 years for confirmation that the rocket had arrived! An astronaut in the rocket will observe length contraction of the distance between Earth and the distant planet, so the distance from Earth to the distant planet in the astronaut's reference frame is 5.95 light years. So the trip will take 7.93 years from the astronaut's reference frame.