I'm working towards my VFR license bit by bit right now. My instructor (happens to be my 2nd cousin) is a commercial pilot though, so he's usually gone and too busy. It's ok though, because I'm in no hurry and usually busy myself. I'm not trying to go commercial; purely for fun and recreation.

One of these days, when I finally do finish the VFR, I'm going to build a 1:1 replica WWII fighter. It will probably be a FW-190! I would honestly love a 109 (more), but I'm
a bit scared of them. They are extremely light, and have that big heavy engine up front (too top heavy). Then the front wheels are FAR too close together. Makes for a dangerous plane to land. During the war, many a skilled pilot got hurt or even broke his back from trying to land one. It's a phenomenal aircraft, and was a spectacular warbird, but I think ol' Willy Messerschmitt basically forgot that the thing had to land until the design was done!

The FockeWulf is much different. Has a wide undercarriage, strong and robust gear and it's very, very rugged. Plus it's faster and (generally) considered superior. So that's probably what I'll end up building. Plus I have the added benefit of accessing an airworthy resto to get it right. *drops coins in piggy bank*
You can get a Cessna or another light aircraft for pretty cheap, really. You can get a used one which needs a bit of work and fix it up for about the price of a used SUV/Cadillac ($20K-$50K). Not bad, when you think about it. Not long ago, I heard some guy near our Metro airport was selling a used 2000-ish Bonanza for about $12K. It wasn't fully airworthy, but you could put another $5-$10K in it and it would be like brand new. So you can get a fix'er-up and make it airworthy for about the price of a nice car if you scope out a good deal. So your dream definitely isn't unreasonable!

I say go for it! You just better have somewhere to store it, or be prepared to pay for storage.
Oh yeah... Surprisingly, a WWII-era aircraft could indeed survive on today's battlefield is employed correctly. You're forgetting, we still use prop-driven aircraft and even slower helicopters!

The C-130 Transport, for one, flies about 350 | 370-ish mph. The 190 could fly flat at 422 mph. With the massive armaments the Germans slapped in those things, you could feasibly use one to knock out C-130s!

P-51 Mustangs were employed in both Korea and Vietnam, right next to jets, missiles and the beginnings of modern aviation technology, and did
very well. The military also uses Cessnas and other very slow prop-driven aircraft regularly. So they
could work. It would just be silly to risk losing aircraft of historical value or spending EXTRA money to build them to WWII-era standards when modern manufacturing techniques could be cheaper and more productive. Otherwise, they could make some decent gun platforms for special roles, like strafing enemy fortifications/vehicles/soldiers or even knocking out slow flying planes and choppers. But money talks and "cool" walks.

There are two major products that come out of Berkeley: LSD and UNIX. We don't believe this to be a coincidence. - Jeremy S. Anderson