Watch the Upper Leading Tones online guitar lesson by John Goldsby from Upright Bass Handbook
Most of these notes are in the half position, the lowest position on the bass. But for some of the notes, like the Db in bar 8 and all of bars 9 and 10 of the first chorus, you'll have to move your left hand up a half step, into the first position.
You should keep your left hand relaxed and you'll be able to reach all of the notes without moving around very much. Just be aware of which position you're working in, half position or first position. You can look at the position of your left hand, and also listen to the way the notes flow. The coordination between the left and right hand is an important factor in creating the groove, and accurately nailing the notes.