Russia and Alaska (according to Google earth) are less than 100km apart, I understand that this part of Russia is extremely remote and the sea pretty rough but on a clear day you see at the very least the islands between the two.
How did nobody at any point in history prior to columbus think to sail across?
As noted, Russia didn't conquer the Far East until long after Columbus had landed, and Chukotka, the region that is closest to Alaska only came into contact with Russia around the 1600s (the first mention of the indigenous Chukchi people by the Cossacks was in 1641). Russia really only took control of the region by the late 1700s, roughly around the same time that James Cook was first sailing up the west coast of North America.
Russia did start sailing along it's own Pacific coast, but there are a couple issues here still: one is that without railways, or even a proper road, it was a long ways to get from Russia proper (meaning the European side) to the Far East: it would usually take around a year to get from Moscow to the Pacific Coast, and even longer to get up north to the Chukotka region; treks of two years were not unusual.
Then there is also the matter of having to navigate the ocean, and that was not easy to do in the Arctic waters separating Russia and Alaska. It is full of ice and very dangerous, and while there was money to be made in fur trading (which was the primary source of income for Russia in the region), it was still not worth the efforts to go that far, when they had literally all of Siberia to trap in.
In short, it was not as easy for Russia to simply discover the New World, in part because by the time they were geographically close to North America, the region had already been explored by Europeans, and so it was largely a moot point.
Well, here's a slight problem. How are you defining 'Russian'? Because while the Russia of today may cover that far, it bears noting that the bits of territory on the Asian side of the Ural Mountains are only recently become part of the Russian state. More can always be said on the matter, so if anyone would like to contribute further as to trans-Bering matters, please go ahead and do so! For the meantime, here are some previous questions on just this topic that you may find interesting: