Run additional wire from junction box to existing light switch?
You may only use wire which is legal for wiring in your country (I am presuming USA and thus National Electrical Code).  The NEC allows many types of wire, but as a practical thing, it boils down to two types:  Multi-conductor cable, typically NM aka Romex; and single-conductor wire used in conduit, typically THWN. 

You are dealing with solid wire, which is stiff.  For a much more flexible wire, stranded wire is available.  However, Romex is not available in stranded.  Each of your devices must be rated for the strandedness of your wire. (for instance "back stab" connections found on very cheap components are rated for solid wire only).  





Neutral must be in the same multi-conductor cable, or the same conduit/raceway, as the other conductors.  You cannot retrofit a neutral to a multi-conductor cable.

If those boxes are connected by conduit, then each of the wires is a single conductor. You can simply pull (or shove) an additional neutral wire through the conduit.

If those boxes are connected by flexible cable, then you will need to pull an entire, additional multiconductor cable between the two boxes.  You have two ways to go.
- Pull a 12/3 or 14/3 cable to replace the existing cable.  This will include hot, switched hot, and neutral.
- Pull an additional 12/2 or 14/2 cable alongside the existing cable.  Cable A will carry always-hot and neutral for the switch and lamp.  Cable B will carry switched-hot and neutral for the lamp. This means when the switch is on, neutral will make a seemingly redundant round-trip between the boxes.  This is required by Code, and for good reason.



You don't need a plumbing snake.  They make "fishing tape" specifically for electricians, used for a variety of hat-tricks to route flexible cable through existing finished work in residences.  Generally in commercial work, everything is in metallic conduit, so any conduit they plan to use must be installed before walls are finished.



If you are starting into electrical work and plan to stay awhile, I'd encourage you early to get acquainted with your local Electrical Supply House.  This is the sort of place that's open from 7am to 6pm and at 7am, the parking lot is full of electricians' trucks.  Electricians are smart people and are perfectly capable of finding the local Home Depot; they buy from electrical supply houses because the pricing is roughly equal and the support and selection is a LOT better.