Can I use "little" meaning the age?
You are still little to do it. Wait until you are 10.
If yes, are there any situations when "little" is preferable over "young"?
"Little" also means "young."
(Of a person) young or younger:
my little brother
Similarly, you can call your older siblings "My big/older brother/sister."
Both, "young" and "little," are correct and can be used interchangeably, but I will be more inclined to choose "younger." Though "little" is more commonly used.
Do not say "small brother/sister," unless you are referring to their physical attribute.
You're just a little kid. You don't know anything is a typical thing to say. Little does not just refer to the size of the child, but to his young age and inexperience. Little refers to everything about the kid being a kid, the child's "kidness".
Oxford English Dictionary:
Of a child or animal: not close to being fully grown, (very) young; (also) younger than a particular relative
Thus, we can also refer to a younger sibling as either my little sister/brother or my kid sister/brother, and they mean the same thing.