Edit: I wrote this answer when I was tired and thought anna was born on the earth not the spaceship, and I don't plan on going back and changing it.
This is kind of a bad question the way you phrased it. Are you sure you're not just interpreting the question wrong?
First, remember that in the spaceship's frame the distance to planet X is length contracted, so it can get there (by its own clock) as fast as it wants. Just to get that out of the way.
You're right in that if we were talking about the Earth's frame of reference and the spaceship's clock, the answer would be no - however rapidly the spaceship travels, it cannot travel faster than c and so when Anna is one year old in the Earth's frame, the spaceship will always only be a bit less than one lightyear away.
However, if we're talking about the spaceship's frame of reference and the spaceship's clock ("spaceship's clock" alone is a bit ambiguous here. We also need to specify the frame of reference), the question changes. The distance between planet X and the spaceship undergoes length contraction so from its frame it gets there faster and faster as its speed increases. Likewise, time goes by more and more slowly on Earth as the spaceship's speed increases (or in spaceship frame of reference terms: as the spaceship stays still and the Earth moves faster and faster away). So those two effects mean that it can get there and as little time as you please has elapsed on Earth. From the spaceship's frame, it can get there well before Anna turns 1.
Is there a contradiction here? No, because the two locations are spacelike and causally separated and saying "x is happening at the same time as y" is relatively meaningless since they're so far away. It's technically correct but physically not too meaningful to say that "The spaceship arrived at the same time as Anna's first birthday in the spaceship's frame of reference", because as you noticed, in Anna's frame of reference the spaceship still has a long way to go (~99 lightyears) when she turns 1.
Try writing out the Lorentz transformation explicitly if it still baffles you.