How to work with an article in a foreign language? After some digging, I found this article from 1969 which was referenced by an earlier publication I was interested in. Unfortunately, it is in Russian, is 13 pages long, and I know of no translation.

This is the first time I have had this problem. In general, if I want to find a way to read the article I am interested in, what should I do? Is it common to ask colleagues to help translate? Are there services which are commonly used? Websites, forums, etc.?
Read it

Get the article machine translated. If you can not copy-paste from the PDF, it would be convenient to do some sort of OCR on it. An alternative that requires work is to retype the (relevant parts of) the article with a Cyrillic keyboard. There almost certainly exists software or internet applications for typing Cyrillic letters. I understand there also exist applications that recognize text from a picture and translate them on the fly.

Have, side-by-side, the machine translated article and the original article. Since you hopefully have domain knowledge, you can read the translation, figure out what the funny and strange parts mean, and get the gist of the meaning. When this does not work, check the relevant passage in the original article and use a dictionary to check individual words. If this is not enough, search for translations of sequences of words. Some of them might be fixed phrases or idioms.

Tatoeba is a fine website for seeing phrases in context and possibly their translations. It also has a non-trivial amount of mathematics sentences in many languages.

Knowing a bit of the language helps

You might want to get familiar with the Russian alphabet and rough pronunciation rules. Get a flashcard program, such as Ankidroid/Anki, and a suitable deck for it, and learn them. It should not take that long. The painful alternative is to have e.g. the Wikipedia page for Russian alphabet open on the side and checking the letters there until you remember them.

For a slightly larger investment, use something like Duolingo to learn the very basics of the language. This, combined with your domain knowledge of mathematics, will make it possible to read mathematical text with reasonable fluency surprisingly fast.

I know a bit of French and can read mathematics in French, Spanish and Portuguese so that I get the main points of the text, and in fact, have a read a couple of French articles in detail. I know Scandinavic languages and can figure out mathematical German, most of the time, without having studied it.

Workload of learning the alphabet

You can spend as much or as little initial time as you want. After the initial rush, this will require an upkeep of couple of minutes per day, and any other activity with the language reduces this to essentially nothing. Source: I have learned hirakana and katakana and upkeep them without learning further Japanese.

Workload of learning the basics of a language

Note that you are mainly interested in passive understanding of written text, which is one of the easier parts. Basics of pronunciation help, because you typically want to read text by pronouncing it mentally at the same time.

Doing as little as a single lesson of Duolingo, little bit of Clozemaster (more advanced) or taking on a relevant Anki deck slowly leads to progress, as long as you have a plan that ensures you learn what you learn fluently and move further every now and then. The daily amount does not matter that much; you want constant exposure to aspects of the language you have not mastered yet and some repetition of the (possibly very limited) aspects you have mastered.

Your objectives are to:


Get a feel for words in the language. This allows you to identify loan words and compound words or equivalent structures. Loan words are frequent and important in technical texts.
Learn a bit of grammar. Where is the verb in the sentence located? What about object and subject? How to recognise a question? 
Frequent words like 'and' and 'or'. Prepositions, even though their precise meaning can be challenging; but at least you will know something is a preposition when you see it, and this already helps a lot.
Along the way, you will learn some vocabulary. Some of it helps (basic verbs, numbers), some not so much (animals, restaurants, etc.). But think of it as a bonus, not the substance.


Something like ten minutes per day, more or less every day, will already lead to slow progress, and every little bit helps much more than it should. This is due to your domain knowledge and knowing a related language (English) or maybe several; most people in world already know at least two languages.

It is a marathon, not a sprint; binging on the language is unlikely to give lasting results. Instead, the aim is to form a lightweight habit.

Verify

If you are relying on a particular lemma from the paper, for example, then it is worthwhile to verify you have understood it precisely correctly. Confusing, for example, "if" and "only if" is not nice in mathematics. But probably your domain knowledge helps you here, so that you will already see if the translation is wrong.