First fix your problem and give the car a tune up before mods. Also make sure that timming belt was done and if you are even the slightest bit unsure do anyway. Also call Nissan to see if your car has had the recall on the injectors done yet. Lastly for mods rather than getting a cat back exhuast you should have it done from the turbo back. Certified muffler has a 3 inch custom down pipe that goes from the turbo down. They also sell a 3 inch custom exhuast system with muffler. Doing a cat back will help over stock but a full exhuast system will be much much better. Other than that a boost controller to run 10psi. After that if you get bored get an intercooler and turn up the bost more. You can also look into a gutted intake manifold with 240sx tb. Thats about as far as my turbo knowledge goes so good luck.
P.S. just like on the S30 the 60mm tb does nothing for performance until you get the intake manifold opened up so hold of doing that until you find someone to gut out your stock manifold. I know there is a guy over on z31.com that is now making them for 500. I you are intersted I'll look up what his name is, can't think of it right now, so you can contact him. Otherwise just hit me up when you are ready. Also don't forget suspension upgrades
It's too bad, that 300zx has way more potential than the 280z. Why did you buy it if you weren't going to put any money into it? You should have just used the 2500 it cost you to save for a rainy day. Also, the only reason your 280z will out run the 300zx turbo is becuase you have way more money into your 280z. If you even put 1/4 of the time and money into the 300zx it would beat your 2+2 with out even trying. It's very possible to get the 300zx turbo into the 13's with only 500 bucks into mods. Then again you have to have a properly working 300zx turbo before you can do that.
Yeah that's the answer; if something is broken bolt on more crap... lol I have friends who think that same way and I personally think it's silly. There's no use modding a broken car; you're better off getting EVERYTHING perfect and THEN upgrading stuff but hey it's your money