Ulan-Ude is the one Russian city where a 42-ton head of Lenin and a six-meter golden Buddha are both local landmarks, and both are loved. It began in 1666 as a Cossack winter post called Udinskoye on the Uda river, grew rich on the tea trade with China, and became the capital of Buryatia, the republic of the Buryats, a Mongolic people who make up a large minority in the city. It was called Verkhneudinsk until 1934, when it was renamed Ulan-Ude, "Red Uda".
Three days is the right length. The city center is small and mostly walkable, so one day covers it. The Buddhist side needs a second day, because the Ivolginsky datsan is a half-day trip on its own. The Old Believers' village of Tarbagatai is the best reason to stay a third day, and it's the sort of place you don't get to by accident. Lake Baikal is only about 100 km west, but in winter the east shore is a serious expedition, not a day out, so it sits in the more-time list with an honest warning.
The plan runs Wednesday to Friday. Start on a Wednesday: the open-air ethnographic museum is listed as closed or half closed early in the week. Ulan-Ude is also the junction where the Trans-Siberian splits, with the branch going south to Mongolia, and the Irkutsk guide is the next stop west. The Russia basics are on the Russia page; the visa, card and phone rules in the tips below matter more than usual, so read them before you book anything.