The transmission casings are identical, so it'll fit directly into the hole & bolt up to everything.
You don't mention if your current box is single range or dual range.
If it's single range, swap over the vacuum lines. The actuators work the same way.
You'll need to install the dual range lever & console into your wagon. They'll bolt directly in.
You can either keep your existing gearstick, or use the RX one with it's switch. The wiring for the RX switch & your existing gearstick switch are the same (whether they simply swap plugs I don't know).
And check the 2 rear diff ratios. Tho you'll want the LSD from the RX anyway
BTW, no such thing as part-time AWD. Its part-time 4wd. Bit of a difference
Does the RX have the skirt kit? I'm searching for the rear side pieces if you're going to be stripping it...