- The goal was his to achieve.
- The goal was for him to achieve.
Both the sentences are correct and mean the same.
The goal was his to achieve.
In this sentence The goal is the subject of the sentence, verb is was and predicate is his to achieve.
Now leave to achieve and the sentence will reduce to The goal was his. His is a possessive pronoun, modifying the goal. The goal is his - this sentence clearly says whose goal it is.
Now add to achieve. The sentence becomes - The goal was his to achieve. Now to achieve part is for his task.
Yes this kind of pattern occurs often and very natural and correct.
Is this clear now?