The Russians would not invade Alaska: their long-term geopolitical goal is to expand to their four coasts: the Pacific (done), the Baltic (done, partially: St. Petersburg is vulnerable to being cut off), the Black Sea (done, partially, no good port in the territory they control: they'd want Crimea), and the Indian Ocean. Alaska does not fit into this goal, which I suspect is part of the reason they were willing to sell it in the first place. And unlike Americans, they think in the very, very long term.
Actually one of the reasons the Russians were willing to sell was that they were concerned that if they went to war with Britain again (as they had just a decade earlier) that the British might just take it without paying for it given how British North America was filling out and becoming a center of British interest again. Anglo-Russian relations have seldom been warm and even then only because they had a common enemy at the time.
