So you’re coming to Lithuania and you don’t know the language. Well, you should be fine as long as you speak either Russian, Polish or English and provided you stay in Vilnius for the pleasures and comfort of English (the further from the capital, the less chance there is).
From my modest experience, Russian is always your best bet. In my everyday encounters with all sorts of linguistically unidentified individuals, I don’t even ask any more and try my luck in Russian directly. The vast majority of Lithuanians, both young and old, speak it to some degree, the probability decreasing with age.
The same goes for Polish, although you will often be surprised to meet young people speaking very good Polish. Don’t expect it to be the standard Polish though, Poles living in Lithuania speak the Northern Kresy dialect, perfectly understandable but with some major differences in pronunciation and vocabulary. If you speak Polish (and no Russian), go ahead and try it, usually people understand it or will find someone who does. In Vilnius, there is always a Polish speaker somewhere nearby.
Now English, naturally, works the other way round. The younger the better. As usual, you will find English speakers in shops and restaurants in the centre of Vilnius as they cater for tourists.
There is, of course, a second, more adventurous option: learn it! There are several schools teaching Lithuanian to foreigners in Vilnius. Here is some of them: The Department of Lithuanian Studies at Vilnius University offers daytime and evening classes on all levels, Lingua Lituanica has various groups and offers individual courses, also via Skype. Center Plus is an NGO offering cheap Lithuanian courses in Russian or English. For more info contact firstname.lastname@example.org. For a completely free online course try Oneness or search youtube. For more fun, try language exchange, ask around the always-active crowd of the Foreigners in Vilnius Facebook forum.
Last but not least, even if you decide you don’t need Lithuanian to be happy, try learning a couple of greetings and everyday phrases anyway. It can work miracles.