The engine swap you are wanting to do will rip up that crappy old transmission.
You'll need to install new mounts.
You'll need a transmission that can handle the power.
You'll need to find a way to mate the new engine to the new transmission.
You'll need to find a way to mount the transmission to the driveshaft.
Come to think of it, you'll probably need a new driveshaft all together.

The ECU from the original vehicle the engine came from will work, so thats solved.

An engine swap isn't really an engine swap. Its more building a car around a different powertrain.

I would suggest just buying a car which already has decent power and upgrading that with easier things like k&n intake, cams, valve springs, rockers, custom exhaust and headers, engine map, blah blah blah.
All that stuff is ceaper and easier than putting a big engine in a crappy car which isn't built to take power.

But even less extreme mods can hurt a vehicle.
I had a 2.5 inch pulley on my supercharger which gave me 20lbs of boost in an engine designed to take 10-12lbs.
Blew my head gasket twice before I changed to a larger pulley. I'm getting much less power now, but I'm not using it on a track, so why kill my wallet with engine problems for 380hp when it takes much less maintenance to settle for 350?

In short,
You could spend 20 grand on a riceburner tuner and it will keep up with modern mid level sports cars. Or you could spend 15 grand on a 350z which is already fast enough to make those $80,000 mercedies look silly.