If it's an 89 you should have no problems. An easy way to tell is if it has round motor mounts its from 89+.
The harness on the 1JZ should hook right up to the interior harness. The only problem is that you'll have to lengthen the harness by about 18".
The only reason you'd need to switch a your crossmember is if they are, for some reason, square. But they shouldn't be. And if they are you can just get
these. Look for the "1JZ Motor mount adapter brackets." Also if you just buy the engine by itself, you'll also need an R154 transmission. With that change, you'll need an R154 driveshaft, R154 mounts,
1JZ bellhousing, and a
1JZ flywheel. Also to complete the swap you'll need
this and
this if you are using the hydrofans. There's a few more things but that's most of it.