How do I replace a sink drain body when there isn't a trap?
If there were a trap there, it would have the same type of threaded fitting that your existing drain assembly has. 


cut the tailpiece a bit shorter
install a 1 1/2" slip-joint nut and 1 1/2" x 1 1/4" reducing slip-joint washer onto the tailpiece
insert the assembly all the way up through the hole in the sink
screw the upper drain assembly flange, packed with plumber's putty, onto the top of the drain assembly
lower the assembly such that the tailpiece enters the top of the drain pipe (it looks like it will line up)
tighten the lower lock-nut, such that it pushes the tapered gasket up against and centered on the hole in the sink, from below. This will tighten the drain assembly to the sink, effecting a seal at the top with the plumber's putty under the upper drain assembly flange. Wipe off excess putty that squeezes out.
slide the slip-joint washer and nut down to the threaded drain pipe fitting and tighten firmly (but carefully) with a wrench. P.S. hand tight is not good enough




picture from Google Images

p.s.s.- I am dubious of your presumption that a trap exists someplace downstream