The closest translation of no problem in Russian is:
– Без пробле́м.
(Lit.: Without problems)
But you can also use some other options. For example, if someone accidentally pushed you and is saying sorry, you can reply:
– Ничего́ стра́шного.
No problem. (Lit.: Nothing terrible.)
If someone asks you to do something and you are ready to do that, you can say:
– Не вопро́с.
No problem. (Lit.: Not a question.)
Listen to "No problem" in Russian
The audio recording includes all the examples (in bold and blue) listed above.