I am English and I purchased a property in Nikolaev in September 2017. I paid $45k for a 85sqm apartment in a newish development called Riviera...
https://rivieracity.com.ua/
I didn't purchase it new; it had already been completed by the previous owners. As you mentioned, new apartments are just the walls and the windows - everything else needs to be completed by the purchaser. Our apartment had been completed a couple of years earlier, so it is not that old now.
We found the property using the Ukrainian equivalent of RightMove, which is Dom Ria...
https://dom.ria.com/
I had a budget of $50k at the time, which gave me a lot of options in Nikolaev. You have about $5k more than I did, so you should also find plenty of options in Nikolaev too. I don't know about other cities, but I think Odessa would be more expensive and Kherson might be a bit cheaper again.
With regards to the process of buying, I would recommend that you find someone who can help you. I am (now) married to a Ukrainian, so it was fairly easy and quick for me to complete the purchase here.