Get the right size cone wrench. As already mentioned, what you really need it for is putting the parts back together.
The right hand side cone needs to be locked tightly against the locknut. It is hard/impossible to do this without a cone wrench.
If the right hand cone is not locked up tightly, it can rotate as the wheel turns, tightening itself against the bearings. You run the risk of locking up the wheel and breaking the axle.
The left hand cone, if it becomes loose as you ride, will rotate in a direction away from the bearings giving you a sloppy adjustment, but not a sudden lockup of the wheel.
Get the right size cone wrench, lock up the right hand cone, put the hub back together, and make the bearing adjustments from the left hand side.