A big part of being a bass player is developing fast ears.
Yup. I think the best thing you can do to learn tunes is not to learn them from reading out of a book, but by listening. Try playing along with recordings of songs you don't know. Figure out the changes (and melody) by ear. This is good practice for learning on the gig. If you are having trouble, maybe check how you did with a fake book, but remember that not everyone plays exactly the same changes. Fake books have their place, but I didn't start really learning tunes until I started leaving the fake books at home.
Learning by ear has many advantages:
1. You are using your ears! Isn't that what we are supposed to be doing when we are playing jazz? Shifting your focus from visual to aural cues helps your jazz playing in just about every way and will constantly improve your ear training.
2. It makes it easier to learn new tunes (eventually). You will start hearing similarities in different tunes making it easier for you to "guess" the changes even on a tune you have never played.
3. You will be able to play in more keys. If you focus not on specific note names, but chord functions (so instead of seeing|Cmaj7|B-7b5|E7|A-7|, for example, you are thinking "root, then 2-5-1 to the relative minor.") it will be much easier to play a tune in any key. This is hard to do when you are staring at specific chords on a piece of paper.
4. You will be better able to adjust when people play alternate changes. I've noticed that musicians reading from fake books are much less likely to be able (or willing) to follow musicians that play the changes somewhat differently. If you are focused on listening and not reading, you will definitely notice when the chord you play doesn't fit in. And with all of your practice in listening, you will be much more likely to be able to figure out on the spot what the other musicians are playing.
I'm sure there is plenty that I am missing in addition to all of this. I also don't want to imply that reading is unimportant, because I think the ability to read music well is also incredibly important. There are many times when we have to sight read music on the gig, so being prepared for this is important. But even here, this experience using ones ears is helpful. If you get lost while sight reading, it is much easier to find your place again if you are listening to the other musicians and can figure out the changes from what they are playing.