I ship to Russia fairly frequently. While it appears that most of you have had better luck with Express mail, I have found standard Priority (not Flat Rate) to be the most reliable followed by First Class. Standard Priority and Express are trackable and insurable up to $5000 (for parcels containing merchandise) according to USPS.com ->
http://pe.usps.com/text/imm/ps_013.htm (this link contains the current mailing conditions to Russia) I do NOT use First Class on most international shipments that I can't afford to lose and replace as First Class is not trackable** or insurable (through USPS) to most destinations to my knowledge.
Some interesting, maybe helpful tidbits -
A large, online, third party insurance company no longer provides coverage to the Russian Federation (tracked or not). They had provided coverage previously.
I currently have two USPS insurance claims out for parcels that went to Russia. I started an inquiry for a third one yesterday. I may be starting a fourth inquiry soon. ALL went Express. It is my experience that Russia is, bar none, the single worst country to ship to with regards to time frames and safe arrival. This pertains to Russian customs and the Russian postal service, not the customers.
**FWIW, there are a few countries that do have Delivery Confirmation using First Class and Priority Flat Rate (envelopes and small boxes)**. I believe this is the most current list -> Canada, Australia, Belgium, Brazil, Croatia, Denmark, France, Germany, Great Britain, Israel, Netherlands, New Zealand, Spain, and Switzerland. I have used it successfully to most of these countries.
**You MUST purchase AND print the shipping labels online! Here are the rules ->
http://pe.usps.com/text/imm/immc2_022.htm
**Purchasing First Class International postage at your post office will NOT give you Delivery Confirmation and is typically more expensive and time consuming.
Hope it helps,
Rob