The Mongol Empire didn't extend quite far north enough to make crossing the Bering Strait very possible even if they would have had the desire to (and why would they?)

At the time of Columbus's expedition, there was no "Russia" as we know it. The entity that would become Russia was the Grand Duchy of Moscow, and the Duchy's territory - while significant - was in the Western part of Russia. The Duchy would become the Tsardom of Russia and continue it's expansion, but the Far East wasn't added until the 17th Century, as shown in this map, well after the time of Columbus. Russia did try to get Alaska, and succeeded - Alaska was a part of the Russian Empire until they sold it to America in 1867.