Kevin, a stud finder, available from hardware stores for about $20, will pinpoint the stud locations perfectly. If you just can't justify the expense, you can calculate the rough position on the ceiling where you want to mount the fixture, drill a small hole (3/8" is about right), then take a bent coathanger wire and fish around until you get a pretty good idea of the location of the joists and the direction they are running. When you find one edge, the next stud will be 14-1/2" away. You can pinpoint the edge exactly by pushing an ice pick, scratch awl or long, thin nail thru the drywall at the estimated stud edge position in several locations until you find it exactly. Then take spackle and fill the holes.
But are you sure that the light fixture you purchased is correct for your intended usage? A 4' x 2' light is the correct size to fit in a suspended ceiling. You obviously can not mount it between the joists. If it is a surface mount and you just need to mount it to the joists, then you are OK.