Which languages to brush up on for Ukraine trip?

Don't focus on either language because you are unlikely to achieve reasonable proficiency in either considering a few hours per week for a few weeks. What you should focus on is reading, especially place names.

Both the Russian and Ukrainian alphabets are similar to each other and not very different from the Latin script that you are familiar with. Learn the letters and learn spellings of the place names where you will be visiting. Learn also to read simple signs such as "restroom", "entrance", and "exit".

The ability to read the signs will bring you immense enjoyment and let you feel that you are in a place where you can experience the reality of the place without needing a translator for everything. It really removes a layer of insulation between you and the place, and puts a small part of the place in you.

EDIT: You might want to employ Anki and Ankidroid to help you study. Anki completely changed the way that I learn languages and other subjects. The truth is, with Anki you just might gain reasonable proficiency in the few weeks that you have.

Russian, no contest. Not quite where you mention (presumably thankfully!) but not far off:

The Russian language in Ukraine is the most common first language in the Donbass and Crimea regions, and the predominant language in large cities in the East and South of the country.

In Ukraine the ratio may be nearer 50/50 but there are approximately five times as many Russian speakers as Ukrainian, so could be more likely to be useful later.

From here, p85:

While there is greater freedom in public encounters, in many contexts there is pressure to adhere to a particular language. In Kyiv, the country's capital, Ukrainian is the language of written official communications, public political communications, and formal interactions in institutions such as schools, courts, and the media; meanwhile Russian is the predominant language of informal written and spoken exchanges, and of economic power (Zazulya, 2003: 138). While the language of the public political sphere is increasingly Ukrainian, the language of business remains predominantly Russian. Ukrainian functions as the language of instruction used during lessons and in official meetings in most Ukrainian schools. However, in most urban schools in central and eastern Ukraine, Russian is the dominant language outside of formal classroom interactions. Schoolchildren generally shift to Russian during breaks and unofficial events. In many cases, principals and teachers do not encourage the pupils to use Ukrainian, try not to pay attention to the language shift, and respond to pupils in Russian as well. Ukrainophone parents in Kyiv have reported that their children feel uncomfortable speaking Ukrainian in school for fear of being ostracized (Bilaniuk, 2005: 4748, 63-64). Children who withstand the pressures of linguistic conformism and use Ukrainian in such russophone language environments are rare. In this context, according to Masenko (2004), Russian can be defined as a language of accommodation (prystosuvannia) while Ukrainian is a language of opposition (protystoyannia). Meanwhile, the prevalence of Russian in informal contexts does not allay the concerns of those russophone parents whose children are taught in Ukrainian and receive little or no formal education in Russian, and as a result, cannot write correctly in Russian.

That research is a little dated but since much is about schooling may relate to those who are now adults.

Being able to convert Cyrillic script into something more familiar could be very useful but that applies to both languages. ????????? may look “Greek” but mere character substitution (to kalendar) would give a good clue to its meaning. The alphabet should be easy to learn in the time available.

Russian and Ukranian are not so very close languages. But the locals used to practice funny type of conversations when one interlocutor speaks Russian (Ukranian) and receives and understands an answer in Ukranian (Russian). May be this not the case now... If I were you, I would concentrate on Cyrillic alphabet, so that I would be able to understand simple written texts in both languages (on signboards, for example). In any case, I do not think it is a good idea for an American to address people in Ukraine in Russian or Ukranian after such a restricted study. You can boldly speak your American English, I think they will understand you better ;-). Good luck!

pnuts' answer is mostly correct but he uses a bit outdated sources.

There were never U/R 50/50 ratio in Ukraine. I would say 70/30 and Russian usage is decreasing since most of the young population learns only Ukrainian.

Unfortunately I have to agree with pnuts that if you want to travel to post soviet countries frequently you should better learn Russian as it is understood in more places.

If your goal is to make an impression on locals I would definitely go with Ukrainian. At the moment Russian language is associated with Russian invasion even though a lot of people still speak Russian. For me speaking Russian in Ukraine is like trying to go to WWII occupied France trying to speak German with french people. I am not telling that you will get in trouble with Russian but you could be more considerate to the local population.

Ukrainian basics are pretty easy and in most places simple phrases like "Thank you", "Please" would be appreciated.

Definitely learn how to transcribe Cyrillic into Latin as it will be very useful for you.

Do stay close with your Ukrainian friend/guide as it is a bit dangerous in the country at the moment. Especially in more eastern regions.

It actually varies by region.

Most people you will encounter will understand Russian but not necessarily respond to you in the same language for various reasons. So depending on where you will be visiting you may need to be able to do both and a few weeks crash course in either one will be insufficient unless you are familiar with other languages in the Slavic group (Czech, Slovak, Polish, etc).

Five to ten hours a week for a for weeks is enough to understand a little bit of the basics of either Ukrainian or Russian. The languages are very similar, so in that sense it doesn't matter too much which language you pick. That said, if you're going to western Ukraine, it'd be more sensible to try your luck with Ukrainian as Russian is more common in the east.

With a few hours of practice, you'll be able to read Cyrillic and with a few hours more, you'll be able to learn a few basic phrases by heart. These together will allow you to read a bit of a menu and order a beer.

With, say, 50 hours, you could manage a few basic interactions. Which is pretty cool, I'd say.

(I use duolingo for learning Portuguese, which is great. Sadly, Russian and Ukrainian are not yet available. It seems they are not... rushin' enough.)

I think learning some courtesies like hello, goodbye, thank you, please etc. does always makes sense when traveling to a foreign country. I am always pleased when I recognize someone visiting my country took effort to learn some basic words in my native language.

Since both, ukranian and russian use the kyrillic alphabet and signs are often written in kyrillic only it would definitely make sense learning the alphabet. In my opinion it always makes sense to catch at least a glimpse of the native language of the country you are going to.