Yeah, "attitude" works perfectly. =)
I think it's even better if you specify the attitude:
--> "Even when she was a child, her attitude towards life had been positive.
Another possibility is to say "...her spirit had been positive".
As for your other suggestions:
"viewpoint", "position" and "approach" might indeed sound strange. The listener will think: What is "a positive approach", because it sounds a little judgemental.
Better version if you want to use this noun: "Even when she was a child her approach to the biggest problems was to solve them all by herself."