Soul wars, does not matter at RS gold all. Range weapon maybe to maintain activity up but if you're on the crowded worlds it does not matter what you do as long as you hit something after every couple of minutes. It'll be the barragers, chinners and 138s using 99 slay that can rule the day. I would just ignore soul wars . Barrows, read a guide. Figure out your method and adjust your gear based on what it urges.
If you can find a Tokkul-Zo, do that rather than an archer's ring: it comes in the elder kiln quest and contains +4 in all combat stats anyhow prayer and strength, which makes it better than the archer's ring. An amulet of which range is probably a better thing to buy than a fury: it has a higher ranged attack bonus although it defends you slightly less well and has no prayer bonus.
You won't need a robin hood hat and an archer helm: to get a quick stat check, the archer helm provides higher defence although the robin hat has +2 more ranged offense. It is really your call, the differences are small. The robin functions nicely as a decorative item and can be recoloured for loyalty points at Xuan in burthorpe and varrock, though. If you don't already have them, you need to get a few barrow's glovesthese items are extremely powerful for your glove slot and extremely helpful consequently. You'll also want to buy a set of ranging boots: you should acquire snakeskin, as ranger boots (12M) don't fall into your budget.
A defense slot will probably involve an illuminated god publication. With Zamorak's publication costing an unholy 22M, a far better alternative is armadyl's book of law in 1.7M: it's just got two less ranged offence, but it also includes defence capabilitiies and as soon as buy OSRS gold you've illuminated it you'll find a great big prayer bonus. Your cape should be ava's unless you are doing something where it will not work, for example shooting ogres in a cage. If that's the situation you should wear your very best ardougne cloak or something else using offensive/prayer bonuses. Red dragonhide, of course, if you don't want to acquire void.