"I managed to finish" is correct and usable in written English. You would use it to express that something was completed despite it being difficult or time-consuming. For example, "I managed to finish my report ahead of the deadline."
I was dying but I managed to finish it".
"I'm really pleased that I managed to finish," Gilmour said.
I managed to finish the game as a striker, which is not too bad.
ALEX KOPPELMAN: I managed to finish my pad see ew a few minutes ago, fortunately.
I'm glad I managed to finish it off because he fought extremely hard".
I managed to finish two 400-page books with airplane mode on before needing to recharge the Paperwhite.
(Still, it managed to finish another run at the Met just this week).
I can now add my dissertation to the list of things I've managed to finish in spite of my less-than-focused approach.
"Hey Boris, I've managed to finish my degree with a first but no husband, can you give me some dating advice?" asked Caitlin Gleeson, while @charcoaltache asked: "I went to uni twice and still don't have a husband.
By the end of the week I'd managed to finish my Zola, had worked my way through my Collected Larkin and completed a decent first draft of half a dozen short stories, which would form the supporting material of my application to UEA.
After a 4-year-long uphill battle and a lot of back-and-forth between the two countries, I eventually managed to finish my medical studies.
I love the desktop app, it’s always running on my Mac. Ludwig is the best English buddy, it answers my 100 queries per day and stays cool.
Cristina Valenza
Retail Lead Linguist @ Apple Inc.