Good info.
I found
this PDF on the Volaris site describing the process and the schedule. Unclear how recent it is. 15 years ago,
someone reported the price to be $15 (answer #5).
Photo
The above photo from 2013 shows a Volaris branded shuttle at Santa Fe Depot; seems like they've been running it for some time, and it looks more professional than I thought.
There also seems to be a
SDMTS Blue-line tram from San Diego to San Ysidro, after which you need to cross the border and get to TIJ. Perhaps there are taxis or combis or buses that can do that short journey.
Hope this helps some people with their research.