Hope you're still planning to embark on this trip.. Sounds like a cracking idea and a lot of fun.
If you haven't locked in your bike choice yet, I'd say DON'T do it on a Postie. Yeah, ok they're cheap (but not really that cheap when you consider the simplicity of components and the fact that you're getting no suspension and no frame for your money). This is not a real bike it's more of a scooter
You will learn a lot about riding and develop techniques for the future years. For this reason you should really get a bike with a proper frame and proper suspension. Optional, but I would also recommend a rider course (going beyond the rider training in the licence) where you can learn some good cornering techniques and then put into practice on the trip. Anyway (sorry Postie fans) the Postie is just not a proper motorcyle and you'll have much more fun, be much more comfortable and develop much better habits/skills on a proper bike.
I've done a lot of adventure riding between Sydney and Coffs Hbr area (2-3 days up, 2-3 days back. Couple pics attached. If you're interested I can give you some info on routes) over the tablelands, through eastern escarpment bush etc spending as much time as poss on dirt but I also enjoy the black-top especially if it's twisty and quiet, and there's a lot of just that in that region. I was typically doing 50/50 dirt/tar and I'd say don't be afraid of the tar, you can have a lot of fun, you can develop motard techniques (leaning bike over more like dirt, you upright) or more traditional road bike cornering (keeping bike more upright), BUT you need a real bike. Obviously you'll have the weight of your gear to carry so you can't go at it like an afternoon trail ride but the skills are the same, it's just a question of a little less right hand.
I also rode up the Cape (the first part - Nth NSW to Cairns - just belting along the highway so can't say much about that). Had a brilliant time from Cairns onward though. There's a bit about it at and a couple of pics in the 'Dorrigo to the Torres Strait' part of this blog Australia to Germany via Asia and the Stans...
I rode an XR600 but I'm not suggesting you need one. There are a couple of different tracks up the Cape (as well as many side detours*) but I found myself in some pretty heavy corrugations and also very deep sand. If you go for the more interesting/adventurous stuff the Postie will be a nightmare on the corrugations with no real suspension and will simply not go through deep sand with a tiny engine and centrifical clutch. You'll be pushing a lot!!
Ok, which bike then? Unless you're really set on the Postie or an Ag my suggestion would be XR250/400 or Suzuki etc equivalent. Honda always has the reputation. As you might read I ran my 600 out of oil but I think I was pretty unlucky as they're supposed to be bullet proof. If you are set on Postie or Ag then I would definitely choose the Ag as they have a lot more real bike/trail bike componentry (esp. new ones with proper shock/swing arm etc), rather than Postie which is like the scooters you see all over SE Asia!
The XR250 would be a great bike. I think the extra (probably only $800-1000) you'd have to put on top of a Postie would be worth it 10 times over. Also, on top of the touring riding, when you get to your campsite you can take all the gear off and you've got a capable dirt bike for an afternoon blatt - think about that, in some absolutely awesome riding locations..
Fuel: I had no problems with 21L (long range Accerbis tank on first bike, 11L tank + 10L canister on 2nd). You will probably even get further with a (more economical) 250 (?)
*You could contact some of the aboriginal communities for permission to access areas on the west coast of the Cape. Not many people get to see these areas. Just needs time and planning.