We use something called ephimerides, which describe how much an object moves in longitude and latitude (Right Ascension and Declination) per day etc. In other words, they describe the trajectory of a moving celestial object. The vast majority of celestial objects don’t move appreciably on the sky from one year to the next, the exceptions being solar system objects (planets, comets, asteroids) and a handful of nearby stars.

>The object we are studying might not move in the same repeatable orbitable pattern around our (0,0)... like say a comet or asteroid that is moving in a straight line out of any given orbit.

Comet and asteroid orbits are just the usual conic sections, most often ellipses. Straight lines would require propulsion; actually, a bona fide "space drive". Once the orbital parameters have been found out, the orbit can be extrapolated for tens, even hundreds of years with some accuracy, and assuming there are no close encounters with massive objects.