Learn how to transpose up a perfect fifth. The perfect fifth consists of seven half steps. With this knowledge you can determine that a perfect fifth above Eb is Bb. While A# is also seven half steps above Bb, you would use A# because A is actually a fourth away from E.
Transpose every note from the euphonium part up a perfect fifth.
Determine whether the transposed part is within the range of the French horn. In most cases, this won't be a problem since the horn can play the entire range of the euphonium. However, if the part uses notes below E two octaves below middle C, you will have to transpose the music up an octave.