Prepare yourself for a long frustrating process. Picking, slap, fingering, etc. are just the technical side of music, the first hurdle to jump while learning an instrument. Learning to play by ear is MUCH harder than learning to produce clean notes. Don't be discouraged if you don't even notice progress. It's over long periods of time that you really notice improvement. Usually I notice that my ear has improved when I play a song that I couldn't transcribe several months ago. Just because it's hard doesn't mean it can't be fun though. When you figure out a song by ear, you own it. I find it much easier to remember songs that I learned by ear rather than from a tab. You will also realize how inaccurate most bass tabs on the internet are.
My advice:
1) Start with easy songs and work your way up gradually. Your technique is probably much more developed than your ear, so these songs may seem boring. But if you pick music that matches your level of technique, you will probably never transcribe it right. That was the case with me for a long time.
2) Try not to hunt around for notes. Learn your theory and try to figure out the song without touching your instrument. Learn to sing all 12 intervals and be able to recognize them (I can't but I'm working on it).
3) Also, sing the part you are trying to play. If you can't sing it, you aren't hearing all the notes properly.
4) Just transcribe something every day. Just like anything else on the bass, you need to practice to become proficient at transcribing by ear. Also, it doesn't have to be a bassline. Try transcribing some horn parts or something a little more melodic once in a while.
Good luck, hope that helps.