More examples of Bond vs soviet agents:
The Spy Who Loved Me - Bond was chased by Soviets (remember that cool ski chase scene?), and killed the husband of the soviet spy referred to in the title.
A View to a Kill: Bond flees from Soviet troops.
According to web sources, in the books that the Bond movies were roughly based on, SMERSH, a Soviet agency, was Bond's primary enemy, rather than SPECTRA organization used in its place in the films. Of course, most of us never read the books, so it is even less likely a Russian would have, while the music only occurred in the films, so maybe that is too tenuous a connection.
Regardless, I assume that most Russians were aware that the Soviet Union and Britain, including their spy agencies, were in conflict throughout the cold war. Surely a cold war generation Russian might have some negative thoughts about a cold war British spy, even in the movies?
It is obvious that several Olympic skaters have decided there is no major negative connotation to Bond music. For that matter, the opening ceremonies in the 2012 Summer Olympics included a Bond movie actor.