It sounds like having your evil moon be an evil sibling planet instead would solve some of your orbital mechanics.
Problems with your moon
1. appearing at night
In order for a moon to only appear at night, it would need to constantly be on the opposite side of the sun, which would preclude it from orbiting the planet.
Note: You could account for this by having it orbit in the opposite direction of the planet's rotation in a geosynchronous orbit (not to be confused with a geostationary orbit) such that it appears in the same position at the same time each day for a ground based observer, and then limit your narrative only to people who live in an area on the planet where the moon is visible at night. (People on the other side of the planet would see the moon during the day, but perhaps they're irrelevant to your story.)
2. long orbital period
If you were to have a moon orbiting a planet with a period of a thousand years, consider that it would need to be extremely distant from the planet-- either all the time, in the case of a nearly circular orbit, or most of the time, in the case of an elliptical orbit. Making that work with a tertiary orbital body (that is, a moon orbiting a planet that is in turn orbiting a sun) is very difficult.
How a sibling planet would work
You still can't have another planet only appear at night, since the farther out you go from the sun, the longer a body will take to orbit the sun. Eventually any planet orbiting closer to the sun will "lap" an orbit farther out; which means that at some point, the sun and the other planet are guaranteed to be in the same general direction (from the first planet's perspective).
In order for a planet to appear on the nighttime side of another planet for 1000 years, the inner planet would need to "lap" the outer planet no more quickly than every 2000 years. Assuming concentricity, the evil sibling planet will necessarily spend more than 1000 years on the daytime side.
Keep in mind that the picture above models a gradual desynchronization... The inner planet revolves around the sun 1000 times with the outer planet struggling to keep up, before finally, on the 1000th circuit, the outer planet passes from the inner planet's nighttime side to its daytime side.
Outstanding problems with the sibling planet approach
The biggest problem I can see is that the evil sibling planet would have the greatest gravitational effect during the 1000 years when it's visible at night.
During the 1000+ years when it's visible during the daytime, it will be more distant from the planet (at times it'll be on the complete opposite side of the sun) and while its gravitational influence will be cumulative with that of the sun, it will also be relatively minor due to the distance.
If the appearance of the planet during the day doesn't actually foretell any astrophysical destruction, and it's just idle superstition, then this approach works fine.
Potential reconciliation: Have the evil sibling planet wreak havoc at night
If you flip the scenario so that the evil planet appears at night and then causes destruction, it resolves a few problems.
First, with a concentric orbit, it can spend 1000 years harmlessly on the daytime side of the earth analog, then spend less than 1000 years buzzing by in a closer orbit.
Second, its greater proximity during the night period facilitates greater tidal extremes that can cause destruction befitting your plot device.
The closer you make the two planet's orbits, the shorter the period of time during which the evil sibling will appear at night and the greater you'll make the gravitational effects.
Sometimes these things are easier to model in two dimensions.