Can I use Lucas oil colors for painting plastic kits?
You can try, but first you will want to degrease the plastic with a little rubbing alcohol (and prime it). The paint will not dry quickly unless you add a siccative like China dryer to the oil to thin it out.

That said, I doubt that your airbrush is designed for oil paints. If you do want to risk ruining and clogging it constantly, make sure to really make sure that the thinned paint is free of clots by passing it through a fine meshed filter. You MUST have a moisture trap installed, because otherwise you will be mixing moisture from the air into your oil - which you definitely do not want. Clean everything with acetone and wear a mask.

But really, that is not the right kind of paint. There are liquid enamels that are better suited to permanent applications, which you can probably find in the store where you got the model. But as Walrus notes in the comments, water based paints are the better choice.