Hi mate, been there done that in 2006.
Basically we had to buy a 1993 Ural Tourist from some shady doode for 1500AU in Moscow. The registration is like this: you go in together and get temporary transit plates, then you go to the special inspection place and they deregister your bike and issue a new registration under your name (new plates).
I don't think its possible to get it registered under your name unless u have Russian residency. I used my dad for this as he lives there.
What you need to do is find someone who will write a 'doverenost' which is a trust paper. Its basicly a piese of paper saying that he alows you to drive teh bike. The paper is valid for 2 years which is more than enough for your needs. Some will tell you that u need to get this paper certified by a lawyer in order for you to take it out of the country, thats absoulute crap. In actual fact no one ever even looked at the papers.
What you need is that trust paper (that you both sign, you buy blanks from any parts shop) and a Tec passport (little card with all the vins and details of bike on it). Just make sure you have those! Also you need a green card (international insurance), that can be done on your name, we used a government Insurance agency, usualy they are located near border crossings. But to be honest no one ever cheked ours ever.
Some sellers will tell you they wont sell on 'doverenost' bacause they dont want to keep paying tax. Im pretty sure they can say that it no longer exists or whatever.
Me and my mate rode our Ural from Mosocow to Finland, Estonia, Latvia, Lithuania, Kaliningrad (Russia), Poland, Germany, Holand, Belgium, France, Spain, Portugal and to Switzerland.
The biggest breakdown was in Estonia. The gearbox outbut seal was leaky and we got slak and dint top it up. So the main bearing got trashed which made the bike unridable. We got lucky and got helped out by local Bikies, who fixed our bike and housed us for FREE!
Other small breakdown included steretched pushrods, which resulted in valves no closing al the way thus making starting imposible but once started ran fine. Easy fix, just adjust the tapets clearance.
Urals have dodgy conectors, so you get a blown fuse every now n then. Solution is to buy a stik of cilicone and seal all the conectors with it, fixed.
Your clutch cable will snap! Good news is there is a station on Moscow metro called 'Sakolniki' there you will find all the parts DIRT cheap. Buy spare bearings, tubes, CDI ignition kit, cables, spark plugs, oil etc. You will need it. Dont mean to scare you but If you have goten this far im sure u know how to turn a spaner. Fixing Ural is easy and fun!
Saying all that you could probaly buy a decent bike in either Estonia, Latvia or Lithuania they have much more relaxed laws.
Heres some pics of our adventure:
Euro trip 2006 - a set on Flickr
You will love the sound of Ural humming away on the hiway, and everytime you breakdown you will meet locals and will have a real adventure for sure.
Hope this Helps.