Because of the noise, she became startled/surprised.
Would it be something like
おとから、かのじょはおどろきになった.? Where I use
から for because.
No, you can't compose it like that. Since おと itself is a mere object and not
an event or nature of something, the combination with から doesn't mean reason.
Instead, it should be おと（の せい）で びっくりした.
Noun + から (not a real sentence) can be used to show the noun is the cause of basis for what comes after, used in sentences like:
ちょっとした不注意から大きな事故になることも少なくない。(big accidents are often caused by just a little carelessness)( example taken fromニューアプローチ中上級日本語 p165)
So it is used for cause and effect, but more for showing the basis of judgement, or describing phenomena in general. I think for a simple kind of 'she was surprised by the noise/because of the noise' type situation it may be more natural to use で instead