The phrase "achieved a reputation" is correct and usable in written English.
You can use it when discussing how someone or something has gained a certain level of recognition or status over time.
Example: "Through years of hard work and dedication, she achieved a reputation as one of the leading experts in her field."
Alternatives: "earned a reputation" or "established a reputation."
Quickly, he achieved a reputation as a tireless advocate for radical but peaceful change.
He achieved a reputation as a brilliant concert pianist beginning in London in 1901.
This is because it has achieved a reputation for being fair and efficient, 80% of Australian CSA clients declare themselves either "satisfied" or "very satisfied" with the service.
He has, however, achieved a reputation as a superb classical dancer, first with the Hamburg Ballet, and then at the Dresden SemperOper, where he is currently a principal.
Software programs like Napster and Gnutella have achieved a reputation for fostering a communal spirit of sharing among millions of music-loving computer users.
Kavanagh achieved a reputation in Ireland and abroad not only for his stark anti-pastoral poetry but also for his criticism of his contemporaries and of the church.
This show covers the artist's peak years, when he was already achieving a reputation outside of Spain.
John Zogby did not achieve a reputation for polling prescience by asking Americans whether the woolly mammoth ought to be brought back from extinction by present-day scientists.
On the other hand Chile has followed a more quantity than quality approach achieving a reputation for affordable wines but on average dull compared to Australian or New Zealand wines.
But they were also the servants of far more worldly considerations, such as earning a living or achieving a reputation, and these should never be discounted in any imaginative re-creation of the medieval artist's existence.
Yet by protecting its historic virtues and achieving a reputation as a leafy, folksy, somehow rural yet suburban community, St. James may have become a victim of its own success.
