Okay, take your tune, figure out what degree of the scale it begins on. (You can do this by comparing the tune with the scale that fits it.)
Let's say the tune begins on the fifth degree of the scale (i.e. the fifth note of the scale). Let's say you want to write it down in the key of B minor (as you would see it written for your E flat instrument). Well, the scale of B natural minor (as you would see it in one of your saxophone books) is:
B C# D E F# G A B
and the fifth note of the scale is F#. So that's the first note of the piece (in the imaginary case we started with, that the tune begins on the fifth degree of the scale).
You can keep going by looking at the intervals between the notes in the melody. Just walk yourself through it, note by note. It helps if you know the tune well enough to sing it while you're notating.
It will also help if you check the original sheet music to see if there are any accidentals to watch out for. If it's an accidental in the original, it will be some sort of accidental in your transposed version.
You can ignore the concert key for your purposes.