I tried borsht in one restaurant only to have fish in my mouth for the first time in twenty years. There are Ukrainian dishes you can try, however. Deruny are spicy potato pancakes. I also had delicious vegetarian airline meals on Swiss Air and Lot. I do not eat meat or dairy products.
Many Ukrainian dishes are heavy with meat and dairy products.
I've been to Ukraine twice. My strategy is to cook for myself as much as possible. If you stay in an apartment rather than a hotel room, this is no problem. You can always find grains, beans and cabbage in the markets. Ukrainians seem uninterested in vegetables greener than cabbage, however.