Warping acapellas can be a little difficult because you don't have an obvious rhythmic reference like drums.
Ableton's warping interface is based on placing markers within a recording, and stretching them along a time ruler. You are saying "this part I have marked should be at this point in musical time." So not seconds, but measures and beats and subdivision.
So to take your questions directly:
what does first downbeat mean?
You know how you can count along to this kind of music by going "1, 2, 3, 4, 1, 2, 3, 4" etc? You're counting beats within a measure. A "downbeat" is every time you count "1". So since Ableton Live has its musical time ruler in your clip view, you generally want to get the downbeats lined up correctly. And since the clip starts at the first 1, that's usually a good place to start.
how many beats do you need to count to place the second marker?
There's no clear rule on this. If the whole recording is at a consistent tempo, you really only need to mark the first beat and the last one, and get them in the right places - since audio stretches between markers, everything will stretch and line up correctly. You can check if you've done it right by playing back the clip and turning the metronome on. You should hear the clicks in time with the beat.
But with many clips, it's very hard to tell where the beats are and whether the tempo is shifting, so it's not uncommon to put some other markers in too. Try it, and do whatever you think ends up sounding best.
Is it worth to chop it to get better results?
This is a creative decision. It's up to you. Again, try it and see what you think.
Now, having said all that, it's worth mentioning that this particular acapella is going to be really tricky, because part of why it's cool is that it's got some rhythmic syncopation - it doesn't always fall right on the beats. In particular, the "first" downbeat is before the first word, so if you drop a marker on the first word, that marker should not end up at beat 1. It's right in between beats 1 and 2.
My advice would be to start with the first word that is right on a downbeat. That word is "stitch." Put a warp marker right on the rhythmic part of the word (probably the "it" in the middle) and drag that to 3.1.1 (the first beat on the third measure). Warp the later parts as best you can, probably by finding other words that are right on a downbeat and lining those up with the first beat in some measure, and checking to see that the measures in between are lining up correctly. After you've got all that, you should have a good idea how much stretching is necessary (not much if your project tempo matches the original song tempo) and you can use that as a clue to get the first two measures warped correctly.
One trick I've heard works well is if you have access to an isolated drum track from the same song, warp that on another Audio Track and play it at the same time so you can check that your acapella sounds like the original that you're familiar with.