AFAIK Mr Snowden is not going to get the political asylum, but he can get the temporary one, meant for refugees who can't return home because of danger to their life. However, the difference might not matter, as long as he would at last be able to leave the airport.
His application will be reviewed for up to three months, and if it will be approved, he'll get the refugee status, allowing him to live and work in Russia for one year. The term can be prolonged each 12 months. If the refugee status will be lost, he will be obliged to leave the country in a month.
With this status Snowden will be able to take his time and even try to gain the Russian citizenship. Theoretically.
According to an international agreement, he will also be protected from the extradition.
Ironically, according to the formal rules, there won't be need for Putin's interference for this procedure, the ministry handles such matters by itself.
We can only hope that Snowden won't mess up our diplomacy anymore.