There are 2 very simple ways to try at fixing mobos which do not require a whole lot of equipment:
1) Baking or Reflowing the board. This can be done in an oven, or by using a heat gun. I have had more luck with a heat gun myself, but it is still no guarantee to fix anything. There are quite a few tutorials on how to do this. Just know that once you bake it once that it will likely need it again periodically, so even if it works you still need to save up for a new board
2) Replacing caps. Bad/bleeding caps are simple to replace and generally do not require a very expensive soldering iron (though the better the iron the easier things will go). Caps generally go fro 50c to $1.50 depending on the quality and size, and you will generally pay for for the shipping than you will for the parts.
Aside from those there simply is not much you can do. You could attempt chip replacement, but you would have to find a working chip and have very nice equipment to remove/install the parts without damaging them, and for the cost of the equipment, and the pain of finding a quality chip supplier, it would be far cheaper to simply replace the board unless this was something you were going to make a career out of.