As you say yourself, "to manage", or "to manage to do something" is the correct translation. For example: "I tried to finish the book, but it was so boring that I couldn't manage." or "I can't manage with all the stuff going on in my life.", or "I won't manage to explain this problem to 10 people in 5 minutes". Also in affirmative: "I managed to finish my work before the deadline", or "Most days I manage to stick to my diet".
There's a similar but slightly more enthusiastic word, "succeed", which is used similarly, but it's better suited for situations in which there are expected difficulties, or a very low chance of success, like "Nobody expected him to finish on time, but he succeeded"
Success comes from succeed, the same way that успех comes from успеть.