The biggest problem with language will be where you must speak with elder people. In some situation you must know Ukrainian language, because there is a big probability percent that people misunderstand you. Which situation - read below.
Places where you need to know Ukrainian language
1. Railway and bus station.
If something happen, you need to go to police. Our policemans even attend the courses of English, but... I saw rarely some program on TV, where journalist check how policeman know English... And results of their level of English are so bad for you guys, so I think you shoud know Ukrainian language. At least to explain what happened.
If you need medical help, you must explaine what happend, where you feel the pain or something else. Our doctors mostly don't know English.
Places where you maybe need to know Ukrainian language
Women in the post always speak just few words, so you don't need to be expert in language, but I think they just don't know English. So maybe you can be quiet and they write everythig in your eyes, but sometimes you need to fill some form, so you can ask someone to help you. I think, your visit to post will be successful even if you don't know language.
There is a lot of young people working on the bank and if you visit big bank, not small branch, there will be people who can speak with you and solve your problem. Why it's in my "Maybe" list. Because if you are not at the center of town and there is only small branches of bank, I don't know if there will ne English-speaking manager.
Waitress inmost restaurant are young, so there will be hih percent of English-speaking people. If you are in touristic cirt, they oftenly have English menu. There will be not a big problem to order something at the restaurant.
What can I suggest you - I always do this in other countries: you can go to supermarket and you can take everything there. Just look a the price. Then you look at the total amount, or give just more money and cashier will give you a rest. So you need just to know a little language, to recognize some product: is it milk, juice or oil.
Where you don't need to know languages? In airport and mostly at hotels.
Of course everything depends on the city, region of Ukraine and people who you speak with. All ukrainians study English at school so if you really need someones help or want to ask something, ask young people and soon you will find person, who can give you answer at least by basic English phrases.