I am doing the trans-mongolia around the same period too!!!
I will be reaching St Peterburg on 5 July, leaving to Moscow on 7th July, then 11 July to Yekaterinburg, then stay short 1 night, 12th July to Irkust, 3 nights, then go to Ulan Ude, and turn into Mongolia from there. So far i only fix my St Peterburg & Moscow schedule caused i have already booked the tickets. Otherwise m still planning on the trip! Maybe we can meet at some points?