If you are here looking for information about crossing the UB from UU, then check out this other website and spare yourself my emotional roller coaster.
The horror of staying close to 24 hours on the train coupled with the high price, made me decide to take a bus over instead. With the help of my Ulan Ude Hostel owner, I was able to get my hands on a Ticket to Ulan Bator for 1500 rubles (no commission, nothing).
The plan was to stay 2 nights in Ulan Ude and then catch the 730am bus to Ulan Bator. Anna from the hostel indicated to me roughly where I should go to board the bus and I also went down personally (on the first day) to check the boarding station and route out.