With a bad instrument you would be best of not trying too hard to capture the actual sound of the bass. If you could get your hands on an underwood pickup (not very expensive and most bassist own one) I would go for that for the main sound. They tend to make every bass sound the same which really benefits lesser instruments while taking a lot away from good ones.
Borrow a fresh set of strings if you can as well as that will do a lot for the sound.
try getting a set of Tomastic spirocore for jazz and the like (non bowed applications).