I suppose do eyebrows is a fixed expression, but I’ve failed to find its exact definition.
My vague feeling is that it just means to raise (lift?) eyebrows to demonstrate dissatisfaction Or something like that.
As far as I am aware, “doing my eyebrows” is not a fixed expression.
In that sentence it means I am one of those people whowants to save the world but I am too busy doingtrivial unimportant things.
“doing my eyebrows” could be replaced by any number of other things such as doing my hair, doing my nails, having my legs waxed, having a massage,etc.