IMHO for the Russian Empire to survive serious attempts at reforms must take place prior to the outbreak of the 1905 revolution, quite possibly through the actions of Czar Alexander II assuming that he survives and then continues his reformation of Czarism.
The 1905 revolution more or less destroyed the monarchy's legitimacy (i.e. Bloody Sunday, the violent crushing of the Moscow Uprising, etc.), which in turn was caused by defeats faced by Japan in Manchuria and in the open seas due to the sorry state of the Russian Imperial Army and Navy.
The Imperial Army and Navy had previously been the cornerstone of the Tsarist monarchy, whose defeat in the Russo-Japanese War (and later WWI) propelled the country into open rebellion against the Czar.
Furthermore, the lack of any serious reforms during and after the 1905 revolution only further ended the chances of the Tsarist monarchy surviving while eroding its authority even further.
So for the Russian Empire to survive, earlier reforms have to be passed preferably sometime during the late 1800's and prior to the Russo-Japanese War and quite possibly during the (longer) reign of a surviving Czar Alexander II.
But it is possible, with a POD further back. The the federalization of the Russian Empire (i.e. something like the federalization seen under the early Soviet Union, with significant rights being granted to minority nationalities and ethnic groups) could very well take place as well, which would serve to stabilize the monarchy and would mitigate national independence movements and would decrease clashes by local ethnic groups.
I'm currently in the process of writing an essay examining the necessary changes required to have the Russian Empire survive, and I will discuss the points listed above.