With the Monty Hall problem the host knows which door is the right one and avoids it. This gives the contestant additional information about the set of doors that the person playing russian roulette wouldn't receive.

RR is totally random, the person who first shoots does not have knowledge about the bullet in the chamber. The only new information you have been given is that there wasn't a bullet in that round. Compare this to Monty Hall, the person not only knows that theres a goat behind the unveiled curtain but you also know that this unveiling was purposefully done and will always show a goat. You don't always know the first chamber doesnt have a bullet in it.