If you mean that this moon has one face always facing its star, then just have a look back at Uranus. With an axial tilt of almost 100 degrees, each hemisphere receives constant sunlight for 42 years, then gets 42 years of darkness.
Give your moon a similar axial tilt, and make it precede/recede at the same rate that its planet orbits the star. That way one side will always point to the star.
That side of your moon that faces the star will only get a night when the planet eclipses the star. If you want that to happen daily, you will need a really short orbital period. For reference, the orbital period of Io around Jupiter is close to 42 hours and 30 minutes. You could have a smaller orbital period by having a more massive planet or a closer orbit.
Is this realistic? Yes and no, tending towards the latter. It would be more believable if this is the work of a Kardashev type III civilization. Satellites that orbit too close to their planets tend to get tidally locked in relatively short geological time. On top of that, the precession/recession you need to lock to the star is very extreme, unlikely to happen (if possible at all) and surely not stable for long.