Unless its a high mount stater engine from a pre 1981 car, then it won't fit a 4-cyl T5 without a financial fight.
The whole Fairmont cross-member and engine mounts can indeed be swapped in. I believe the whole engine comes out on the cross member.
Keep the drive shaft, but expect to have to track down one of Big Al's cheap adaptors T5 adaptors and an earlier high mount manual bellhousing if its a high mount engine. You could alternatively try and track down a 78 to 79 T-4 4-speed bellhousing and clutch kit. Many Fairmonts and Mercury Cougars came with four speeds, its just a case of finding one.
I must stresss, you must have a highmount starter 3.3 for this to work.
If its a low mount 1983 block, you'll have a bun fight on your hands. It's not impossible, but just difficult.
For a low mount block, you'd have to grab a Mustang 5.0 bellhousing, and get the clutch plate fitted with the smaller bush which allows the narrower 4-cylinder input shaft fit to the bigger 5.0 clutch.
Mike has a picture of the V8 transmission to low mount starter 200 big bell block in the tech pages.
You'll need to get a blank, zero balance 157 teeth flywheel from Summit, and redrill it to 2.75" .
The V8 low mount bellhousing is meaty, and can be redrilled to fit the low mount block. You have the option of using only the Mustang 5.0 starter, as I think it is 157 teeth. If the auto 3.3 is low mount, it will have a 164 teeth flexplate and starter. If its high mount, it will be 136 teeth, and the starter won't interchage with the 5.0 item.
Lot's of little details.
Don't give up untill you got the information you seek!
