Given his objective at the time was Ecuador, which lies across the sea from Russia, I find it very hard to believe he'd be crossing borders without taking at the very least a plane. And about seeking asylum, you are correct that a passport doesn't stop you from getting asylum, but when the official plane of a head of state is denied entry to the airspace of several European countries due to US suspicion that maybe snowden is on board[0], things become more complicated.

Trying to get himself smuggled out of russia is not realistic, and honestly kind of pointless at this point.

[0]: https://en.wikipedia.org/wiki/Evo_Morales_grounding_incident