Local syncing is good
Some phones have a USB master mode, but not all (as trythis mentioned), and for some of those having it it needs rooting (“jailbreaking”) the phone to activate it. But I like the local syncing idea because it’s so simple. It will only need a piece of electronics that (1) accepts a USB stick as USB client, (2) accepts the phone as USB client and (3) accepts instructions from an app on the phone to copy certain files between these two USB mass storage devices. It will even be faster than via wifi. A Raspberry Pi could do that (it saves the $$ for the wifi module and lowers the power requirements). But I am sure we can find simpler and cheaper devices which can also do this, since it’s a pretty simple task. If @trythis knows the simplest solution, I’m sure he’ll let us know 
Some more ideas that piled up in the last days for gradually connecting these library servers to multi-village intranets and finally to the Internet:
- Connecting villages together with DIY data links. Appropriate tech could be for example the Village Telco kits, the APRS-IS packet radio network, the Guifi DIY long-range wifi network technology, or visible light data connections with cheap laser diodes (which is licence-free electromagnetic spectrum, so everyone can play with it …). Obviously, laser diode connections require line of sight between two villages each, but that is a given in Nepal's hilly region. These connections can easily go from village to village until reaching the next city with "real" Internet access.
- Maybe winged drones (or even turtles) could deliver the USB sticks in a delay-tolerant network? :-)
- This whole idea could very well also become a civil society dialogue space (with a slow but rich dialogue) in rural areas, according to Natalia's comment.