Locating the paper on MathSciNet is always a good idea: http://www.ams.org/mathscinet-getitem?mr=192581
First one should look at electronic mathematical libraries such as http://gen.lib.rus.ec and http://library.nu. Quite often they have an electronic version of the book, which unfortunately is not the case here.
Another efficient method is to look for papers that contain references to the paper under consideration and then ask the author of such a paper for a copy. In our situation a search reveals just one such a paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.9995&rep=rep1&type=pdf
Unfortunately, the author explicitly states that he was not able to locate a copy of the paper that we are looking for.
If all else fails, one should do a search on WorldCat to locate libraries that have a paper copy. In our case a search reveals quite a few libraries: http://www.worldcat.org/search?q=trudy+statistike&dblist=638&fq=yr%3A1960&qt=facet_yr%3A (When looking for a book whose original title is written in a non-Latin alphabet, one should be extremely careful in formulating the request. In our case the words “trudy” and “statistike” seem to be most resistant to distortion by transliteration systems used by different libraries, hence we enter them into our search query.)
Numerous university libraries in the US, France, Germany, Denmark, Slovenia, Czech Republic, Switzerland, Poland, and Australia seem to possess the book you are looking for.
Most probably your university has interlibrary loan agreement with at least one of these universities. Perhaps one of the libraries might be willing to scan the paper for you (maybe for some small fee), you might have a friend in one of these universities, or when traveling to a conference at one of these universities you can simply stop by the library.
Writing directly to Kovalenko (currently he seems to be affiliated with London Metropolitan Univeristy: http://www.londonmet.ac.uk/depts/cctm/staff-pages/staff-a-z.cfm) and requesting a paper copy should also be an option.