Some of it comes down to the technology and resources you have/understand. Figuring out a melody like this might work with two ways, first is humming it. This works more or less reliable with me, at least to create the melody, record it and then figure it, because it is more pure, altough the song is already pretty sparse.
Second thing is (either with humming recording or the actual recording) you can put it in a DAW, and then you have some options. Try to track it with something like melodyne, there's a free trial. (This might even be possible without a DAW, i don't know since I never used melodyne outside a DAW) Melodyne gives you the pitch for the whole sequence you put in, it has a piano roll, and shows you over which notes the melody goes, as well as the timing. This might work on the song, but surely will better work on a acapella version or the humming version.
In a DAW could also set the bpm and try to insert a piano vst, to figure out the melody. The big advantage is, that you are not working against time. Figure out a note that works for most of the melody (key center) and then just align it with the rhytmical pattern. Every note that doesn't fit you just shift till have the whole melody figured.
