Incoming links: Absorbing Markov chain
Since Snakes and Ladders is nothing but a Absorbing Markov chain, the results are exactly the same as for that general problem.
www.jstor.org/stable/3619261: How Long Is a Game of Snakes and Ladders? by Althoen, King and Schilling (1993), paywalled.
"Game" is a bit of a stretch as there are no player choices at all.
A more precise word would be simulation.
More precise, this "game" is exactly an absorbing Markov chain.