I haven't done this route (nor ever met anyone who has) and I include it mostly to answer questions about the Mongolian half of the equation.
This route is accessed from the north of Lake Khubsugul. For Russians and Mongolians, the access won't be too difficult, as you can cross the border at Mondy and head down into Mongolia. There is even a tourist agency that offers this tour for a not-unreasonable price. If you're Russian and interested, let me know and I'll see what I can do.
If you're not Russian and not part of an official diplomatic or scientific expedition, you simply CANNOT cross the border at Mondy. The one road is completely blocked with a guardhouse watched by several border guards armed with automatic weapons and very literally, a license to kill. The route from Russia is easy enough that most people can handle it just fine. It's also a gorgeous and fun route; I highly recommend it.
If you're already planning an expedition through Mongolia and want to add Munku to the itinerary, you can take the normal border crossing at Naushki/Sukhbaator or Kyakhta/Sukhbaator where the Trans-Mongolian and road take nearly all traffic. However, from Ulan-Baatar, it is a several day trip to Lake Khubsugul, and then a very rocky, very slow journey (a long day or two days) along the side of the lake up to its northern end. The trail would take off from here.
I have heard that this route is the easiest, however I have never done it.
The south side of Munku has less snow and a much shorter access route as you don't have to wind your way through river canyons. That said, it may be just a long boulder scramble (probably around 2000-2500 metres' gain) or may involve some technical moments, though I have heard that there are non-technical routes.
If you're really interested in touring through Mongolia, there are tour agencies in Ulan-Baatar who can arrange a trip. In this case talk with them about gear.
A note that Americans do NOT need visas to go to Mongolia. So, find a tour group and head out!