"soon managed" is correct and usable in written English.
You can use it when you want to express that an action or situation was efficiently or quickly handled or sorted out. For example: "We had a problem with the computer, but it was soon managed."
He soon managed to flee the country, taking his family to Paris.
After the departure of the health department, the Beales soon managed to make their house comfortable again.
The new professor did not restrict his activities to the classroom and the concert stage; he soon managed to establish an endowed program named after Duke Ellington.
Quintero, who rescinded his resignation, soon managed to negotiate a reduction in the tariff, and his appeal to Richmond eventually led to Carvajal's arrest.
Activists soon managed to bring sound equipment, powered by generators tucked in the trunk of a car, he said, and he wrote his first song, "Syria Wants Freedom".
But I soon managed to swim with the Straussian flow, and even developed a lasting affection for the stigmatized post-"Rosenkavalier" operas.
Pininfarina was soon managing the relationships with motor manufacturers, and design contracts for Maserati, Alfa Romeo and Peugeot quickly followed.
He likes powerful women.' Kathy Keeton was soon managing the financial side of the business, and they married in 1968.
Success was hers -- she was soon managing construction projects around the United States -- but she had to fight for respect.
He was soon managing a 100-acre orchard, and then a 12-acre farm in Southern California, which grossed close to a million dollars.
"Tall, handsome women soon manage to get home again but the ugly ones stay a long time … some of them, indeed, as much as three or four years".
