If you don't want to take a taxi or Uber, or Yandex.Taxi (which might be a good idea and is also not so much expensive $2-3), you can take bus 119э, which connects the airport, through downtown, to the other side of the city.
Here is the route with stops: https://2gis.ua/kharkov/route...
It goes right via Freedom Square:
I would recommend taking a car, I am taking a taxi when I am heading to or from Kharkov airport. There is no subway route in this part of the city with the airport connection, and public transport there is not good (even though a direct bus connecting your points of interest exists).
Whether you decide to take a bus, or you'd look for a prepaid SIM card for your phone, take a look at this Google Street View picture (the link will take you to Google Maps to the point of interest). It is the bus stop near the terminal, and the building in the background above the bus is the airport terminal itself. This should give the idea about proximity - it's all within a walking distance.
The box between the bus and the bus stop is a 24/7 shop which should have [cheap] prepaid SIM cards - I don't think you can find them inside the terminal.
Welcome to Kharkov!
Taxi update. Taking a taxi is easy when you have a local mobile number. You don't need to speak Russian or Ukrainian, or expect someone to speak English or another language if you take advantage of Yandex.Taxi (to my best knowledge it works well despite the so called sanctions - just make sure to download the mobile app in advance) or Uber. For alternate options of regular taxi services one still needs a phone to call for a car, some popular taxi services to mention - the links lead to mobile app references on their sites, I believe some of them are available in English as well, not just in Russian:
As far as I remember there is some authorized taxi service advertised in the terminal, however I have not heard of it otherwise.
If you don't have a local phone number the options are:
- to purchase a cheap SIM card in a kiosk nearby (e.g. at the bus stop - see above)
- to use free WiFi at the terminal to book a car using mobile application
- to accept a shady taxi driver's offer in the terminal
The airport is located close to urban area but not in the area where taxis pass by otherwise. Shady taxi drivers are not very greedy as alternate car hiring options are simple. There are not so many flights in the airport either, so they know the schedule and the times to wait around to find a passenger in need. They might charge more if they see you have problems calling taxi otherwise, but this should still be more or less reasonable price-wise. I can't think of a taxi driver who refuses to take a passenger to Freedom Square for $10 (as I wrote above regular taxi trip price is $2-3; the fare on public bus is around perhaps $0.10 and can be paid in cash to the driver).