Boy, other than "it depends"? Sheesh! Because, really, it depends! It sounds like there are two topics for you to work on, the upright and jazz, or is it just the upright? Depending on how much time you have for practice, you should probably study the basic techniques of the upright with weekly or every other week lessons for a few months. In my case that lasted 6 months but I would have gone longer if my teacher had been more flexible. Still, I got enough of the basics to practice on my own for a year or so before I needed further instruction to get to the next skill level.
If you need to study jazz yet, I think it's a life-long pursuit but you don't need a teacher after you get started. It'd be useful to study with someone who can show you combo etiquette and walk you through reading Real Book charts and transcribing tunes. They could also practice with you which is useful. That could go on for a year or so, again depending on how quick a study you are, how much time you have to practice, and how much of the work you want to do yourself. Ultimately, IMO, there really aren't any shortcuts; jazz bass requires great ears and that requires consistent focused practice and transcribing.
Okay guys, flame away!