If Putin dies tomorrow the prime minister takes over for "90 days or until an election can occur". However, as I understand it the current Prime minister is considered to have little power or ambition and not the type to run the county long term.
In theory what is suppose to happen next is that the people elect a president they feel will server well, and when his term is up if he fails to live up to that expectation he will be replaced with another president just like any democracy.
In practice Russia hasn't been much of a democracy anywhere but on paper for awhile now. Even when Putin theoretically left office he still ruled the country by proxy and he has made it possible to reset the term limits recently just so he can stay in office until some time in 2030, at which point if he was still in office he no doubt would pass some new legislation saying he can stay in power longer. In short he is straddling the lines between democracy and dictatorship, with some elements of oligarchy sprinkled in, thanks to Putin.
So if Putin suddenly disappears, is there any chance the country could go back to being a democracy it is in paper? or will some new Putin 2.0 take over and continue to rig everything so he can stay in power? If so who is that new Putin replacement likely to be?