The thing about getting a Russian visa in China is that you can apply for a tourist visa if have or had a residency there that's valid for at least 3 months. You will need to get the original visa support mailed to you by post or courier and this can be obtained on http://waytorussia.net/Services/VisaSupport/Tourist.html
If you have not been a resident there you can still apply for a transit visa (maximum 10 days) just with the train tickets (no visa support documents needed). It doesn't give you too many days for stopovers as the Trans-Siberian takes about 4 days to pass through the Russian territory, but if you just want to pass through Russia and stop for a couple of days at Baikal and for a day in Moscow, this could be an option.
If you don't have a residency in China and want to stay in Russia longer than just a few days, then you can get a business visa. Don't be scared off by the name of the visa: it just means that the purpose of your travel should be "business" and not "tourism", but that business could be "exploring opportunities" or "visiting conferences". You can get the invitation needed for this type of visa on http://waytorussia.net/Services/VisaSupport/Tourist.html
This type of visa does not require you to have a residency permit in China to apply, but it costs a bit more and the invitation takes longer to process. You will also need the original of the invitation, and that'll take about 1-2 weeks to process and then you can have it sent to you by courier post.
Alternatively, there are some reports that the rules at the consulate in Shanghai are easier, so you might be able to apply there without the residency permit even for a tourist visa (but it's still better to get the original).
So the best choice is to try to get the Russian visa before at some other country. If you're not sure about your exact dates of travel, you can get the visa valid for the whole month (max period) and then travel any time within this date span.
The problem with this option is that the earliest you can apply is 3 months before traveling, so if, say, you're on a world trip and don't know where you'll end up and when, then it's a bit complicated. In that case, get the business visa valid for 3 months and then you're free to go when you want.
If you're already in China then I'd say try to apply with a copy of the tourist visa support at Shanghai first, if that doesn't work, get the original, try again, if that doesn't work again, get them to send you originals of business visa support and apply with that.
If you don't want to apply to the consulate yourself, you can do it through a local visa agency. You will still need the invitation, which you can get on http://waytorussia.net/Trave/VisaSupport.html
and then once you obtain the invitation you can go to the agency in China listed on this page to have them apply for the Russian visa on your behalf: