One of my favourite fishing spots in all of Australia is north of Broome at a place called Middle Lagoon. It's half way up to Cape Leveque. We had a tinny though. We just sat on the water and listened to where the tuna schools were busting up and headed that way. Heaps of tuna, spanish and even caught black jewfish and coral trout on the troll. You could go there and beg to tag along with someone there who has a boat. Maybe a donation of a case of beer would help.
Further up you can flick lures at Cape Leveque. Tides are big so can be challenging.
Take heaps of lures and wire trace if you get on to someones boat otherwise the trip back to Broome for supplies is a long one. I know, I've done it.
I was there in 2002 & 2004 so a lot could of changed.
A couple of stations on Gibb River Road have places to fish for Barra however it can be hard during the dry.
In Kakadu a lot of people fish for Barra at the causeway at Cahills Crossing however the crocs there can be huge.
You can hire boats and fish from the shore at Dundee Beach near Darwin.
Even the harbour charters in Darwin harbour are ok.
Things I wouldn't go without:
Yabbie traps to catch cherabbin. Drop into any river with dry dog biscuits or leftover meat/bone scraps. They are one of the best tasting seafoods I've come across. They are killer barra bait too.
Insect spray with Deet otherwise you will get smashed by insects.
Epirb - as mentioned before.
Axe/saw for fires if you're bush camping.
Protection for your rods as they will get bounced around. I got a flanalet sheet, folded it several times and got someone to sew the side. Makes seperate pockets for all your rods.
Rod bells so if you are lucky enough to be camping near a river you can set your rod up and sit by the fire waiting for the bells to go off.
A strong spotty so if you do the above you can see the crocs. It always amazed me how many of those beady eyes you can see at night compared to the day time.
Walkie talkies so when you are 2km up the river fishing you don't have to walk back 2km to say your still fishing and haven't been eaten by a croc.
Have a good one.