Define Russian ancestry.
The Israeli Jews who lived in Russia or are descended from such an ancestor are mostly Ashkenazi. Ashkenazi Jews are not ethnic Russians, so if by Russian ancestry you mean Russian genes then no. I'm sure there are Jews with a bit of Russian in them in this way; however, Jews in Russia were very much a tight knit community that didn't marry out much. Russia in particular had the Pale of the Settlement, an area in which Russian Jews *had* to live in, and even many cities within the Pale forbade Jewish residents. Needless to say, the Russians were pretty good at not mixing with the Jews.
However, if by Russian ancestry you mean those who ancestors wandered and settled in Russia and aren't part of the ethnic Russian group...then no. It's estimated that the largest group represented by Israeli Jewry are the Mizrahi Jews, or generally speaking the Jews from Arab/Muslim lands. That's even if by "Russian ancestry" you're not just sticking with modern Russian boundaries but including lands at some point a part of the Russian empire, for example the Ukraine or Belarus. I think groups like Lithuanian and Polish Jews should never be counted as "Russian Jews", but some people do that anyway not knowing the difference so I do bring that up.
Of course, the Mizrahi represent Jews from Morocco all the way to Iran and even beyond, so the comparison parallel is the Ashkenazi. Which Russian Jews are a part of...but do not absolutely dominate. If Ashkenazim outnumber Mizrahim, that still doesn't mean that *most* Jews have Russian ancestry given that Ashkenazim come from communities that sprung up all throughout Europe. When it comes to dividing up Ashkenazi, Mizrahi, Sephardi (yet another division) and the rest into national origin, Russian Jews may be the largest of all of them (given Russia's size) but at that point, it's only the biggest piece of a cake that's been divided a lot. Putting the other pieces together to form a non-Russian piece will show you that, no, most Israelis don't have Russian ancestry.
Basically, no matter how you define "Russian ancestry", the answer is still no. Most Israeli Jews do not have any sort of Russian connection.