Ukrainian and Russian are partially mutual intelligible. I as a native Russian speaker can read Ukrainian and usually understand the most but spoken Ukrainian is relatively hard to understand, although partly it is clear too.
Both Russian and Ukrainian are East Slavic languages. The closest language to Russian is considered Belarusian (also an East Slavic language) and therefore Ukrainian is probably the second closest language.
As far as I know there are lots of native Russian speakers in Ukraine who are usually bilingual and given the history of Ukraine and the current political context it's reasonable to assume that Ukrainians generally tend to know Russian better than Russians know Ukrainian because there's virtually no use in knowing Ukrainian in Russia. However, being fluent in Russian in Ukraine is fairly important, for example, as other answers stated, even current Ukrainian president is bilingual which I think is of use for him.
That being said, I think Ukrainians shouldn't have any problems with acquiring Russian quick enough even those who haven't already learned Russian.
The mutual intelligibility of the Slavic languages (going far beyond the pair Ukrainian and Russian) is a fascinating theme for linguistic research, and there are works out there trying to measure the distances, e.g., in terms of lexical distance.
A more difficult question is the psychological one: Does a potentially traumatized Ukrainian person want to be addressed in Russian? It may be worth trying any other common language than Russian before switching over to Russian as language of intercommunication. Some person may have strong feelings in this respect, others not.
You have 2 questions in 1:
- How mutually intelligible the 2 languages are.
- What's more practical to help refugees.
In terms of mutual intelligibility, it highly depends on context, an educated (the English equivalent would be having studied Chaucer/Shakespeare in English lit in high school, because a lot of Ukranian words are very similar to old Russian words, that aren't in common use anymore, but would be recognizable to someone who'd encountered them in literature) Russian speaker needs to spend a day to learn to listen to vowel shifts and the different inflections + learn ~150 truly different common words to comfortably follow e.g. Ukranian popular political speeches, debates on talk-shows, etc. because those are aimed towards the masses and use simple language as well as using a bunch of the same foreign loan-words(inflation, corruption, computerization, etc.). A high-school dropout in a specialized setting would really struggle. Conversely if he was with Ukranian high school dropouts, he would probably be OK, since the swearing is identical.
In terms of what's practical to learn to serve refugees, definitely Russian. To get a sense of what language people from different regions use day-day, check out Google Trends and compare different Russian vs Ukranian terms for searches within Ukraine. Here are a couple of examples (last 12 months): Kyiv/Kiev: 13% Kyiv/87% Kiev Cheap tickets: 18% Ukrainian Note that Google counts Crimea and the separatist regions as part of Ukraine for this purpose, so accounting for population, add ~4% to the Ukrainian total.
Interestingly enough, even the result for Russian aggression in the last 30 days, is only 39% Ukrainian.
Even Poroshenko and Zelensky are native Russian speakers who learned Ukrainian to enter politics. Here's a video of Zelensky using his gym time to learn Ukrainian, 3 months before he was elected president: https://www.youtube.com/watch?v=PohoNwCDOFk And Poroshenko had several infamous lapses, when he forgot the Ukrainian word for something and had to ask his aides, e.g. https://www.youtube.com/watch?v=Y57R2ldt8P4 He's asking his aide how to say wallet in Ukrainian.
Also note that refugees are more likely to be from the Russian speaking east. Here's today's video of Ukrainian ultra-nationalist politician Iryna Farion complaining that of the Ukrainian refugees in Poland, 80% are speaking Russian in the streets and the Poles are worried that now Putin will have an excuse to invade to protect Russian speakers in Poland. Now, Farion is an extremist and is probably hyperbolizing, but this is an amusing data-point I thought I'd include.
Russian and Ukranian are mutually intelligeable to a significant degree... perhaps to greater extent than, e.g., French and Italian. However, it is necessary to keep in mind that the situation with understanding is not quite symmetric here:
Ukrainians speaking/understanding Russian
Russian was lingua franca in the USSR and before that in the Russian Empire. Thus, everyone in Ukraine, above certain age, speaks perfect Russian. Moreover, Ukraine had been tied to Russia for much longer than other Soviet subjects, and the two cultures intertwine - e.g., Gogol, Shevchenko, and Babel have been considered classical Russian writers and studied in Russian schools (or at least were studied in the Soviet period). Thus, one hardly could expect here a situation where the post-Soviet generation wouldn't speak Russian - as is the case, e.g., in the Baltic countries. Indeed, in post-Soviet Ukraine one could often see television programs where different participants would speak Ukrainian and Russian - depending on their background - and understand each other. Abroad, until recently, Ukrainians would freely mingle with Russians, and largely treated as "Russians" (often the blanket term for anyone coming from the USSR area). Given large Russian-speaking communities in Europe, United States and Israel, as well as in the former parts of the Soviet Union, Ukrainians will certainly find being able to speak Russian helpful. Certainly, some Ukrainians would be too traumatized to speak Russian... but it is more reasonable to expect that Ukrainians would be much more attuned to the difference between Russian, Russians, Russia and Putin than the non-Russian inhabitants of the hosting countries.
Russians speaking/understanding Ukrainian
The situation is not reciprocal, as few Russians, beyond professional linguists, have ever had need to study Ukrainian - even those living in Ukraine could perfectly survive speaking Russian. This was one of the reasons for the resentment against the Ukrainization policy in post-Soviet Ukraine. The political developments of the last decade however have triggered interest in studying the language among the Russian speakers outside Ukraine... notably in Russia.
Nearly all Ukrainians understand and speak Russian fluently, even from the west of the country (Lviv, Ivano-Frankovsk regions). This is not because the languages are similar but because Russian was the state language in the USSR, and remains an important language of international communication (it is second only to English in the Internet for instance).
If you go to Moldova or Kazakhstan, you will find that a lot of people there also speak Russian, despite the fact that their languages are not as closely related to Russian as Ukrainian is.
On the other hand, Russians have troubles understanding Ukrainian. Here is an interview, made about a week ago, of former Polish president Kwasniewski to an Ukrainian journalist Dmitry Gordon (in Russian), where he tells a story (the link is with timecode) of him being with Putin invited to a celebration of a 10 years jubilee of Ukrainian independence in 2001.
Everybody was speaking Ukrainian on the party. He tells that Putin asked him "How much (of Ukrainian) do you understand?". Kwasniewski replied "about 90%". Putin said "but I understand only 30%". Kwasniewski of course could use advantage of knowing both Polish and Russian.