It's possible to swap in a more recent 5 speed. There are even dual range 5 speeds that'll go. It's not exactly a drop in. It'll bolt right onto the engine, but you'll have to modify either the trans mounts or the trans cross member. And your drive shaft won't fit anymore. I can't remember if the donor car will give you the right size shaft or if you have to have yours modified.
As for the engine, you could step up to a SPFI EA81 or an EJ series engine. Both swaps are rather complex as those newer engines are computer controlled so you need the entire harness from the donor car and integrate it into your cars wiring harness. If you're going that far, you might as well go with an EJ as it's a lot more power than the fuel injected EA81 which isn't that much stronger than your carbed EA81. There are people on the board who really know how to do these swaps. There are also many threads with in depth information on all aspects of what you're asking for.
Another option that you don't seem to be considering is modifying your current engine. The primo setup seems to be EA71 pistons, something like .030" shaved off the heads, a Delta high torque custom grind on your cam, and a Webber 2 barrel carb. From what I've read, the whole package is worth 30 to 40 HP. I've only done the carb myself, felt like I gained 5 to 10 HP, but my old carb was jetted way too high, if you're starting with a healthy carb, the change could be larger or smaller.
I'm still rather new myself. But I've been reading tons and trying to figure out just how far I want to go with my own Brat. I'm happy to answer any questions you might have, but I'm far from the best source of info you'll find here.