Is it possible for a moon to continuously have a side facing its star whilst orbiting a planet? For example; is there any possibility for the moon to always have a side facing the Sun whilst orbiting Earth? And if so then what would the day cycle be like?
I worked out the math, it's pretty straight forward.   If we take the Hill sphere, which is an estimate for the furthest possible orbit, and I'm going to just run the math on circular orbits.  Eliptical orbits are harder to put in resonance anyway.

the simple Hill Sphere formula.  $3\frac{r^3}{a^3} = \frac{m}{M}$, where $a$ is the semi major axis planet to sun and $r$ is the distance moon to the planet, little $m$, mass of the planet, big $M$, mass of the star.

To calculate tidal forces, the force changes with the 3rd power of the radius, so for equal tidal force on a moon from both the star it's planet orbits and the planet it orbits, $\frac{r^3}{R^3} = \frac{m}{M}$, and that rather neatly puts the equal tidal forces outside the outer edge of the Hill Sphere by a factor of the cube root of 3, or about 1.44

The outer half of the Hill Sphere isn't long term stable anyway in real systems, so for the Moon to experience a stronger tidal force from the star it's planet orbits than it experiences from it's planet, it would need to be well outside the stable orbital region around the planet, by at least a multiple of 2.88.   

There's no way for any moon in any system to be tidally locked to it's star and not it's planet.