If you're using a smartphone, you can easily install a Ukrainian keyboard.
If you're using a browser, you can use the word bank instead of typing in (there's a switch button at the bottom).
Or if you really want to type without a given choice of words, you can also type using English transliteration. E.g. this exercise would be "ya ne spav". Unfortunately, it's unclear how Duolingo decides to convert Cyrillic letters to Latin ones. Other contributors in this course don't know, contributors from other Duolingo courses don't know :( The safest bet is BGN/PCGN, but it doesn't always work 100%. For example, if you use this online converter you will get "ya ne spav", but in this exercise it will tell you there is a typo. However, I have no idea where it thinks there's a typo :/