Net exports (X - M) is a component of AD, which means that if an economy is a net exports (i.e. they export more oil than they import), then a rise in the price of oil will lead to a rise in X - M and thus AD as well. The reason it should rise, I believe, is because the PED for oil (a commodity) is very inelastic as it doesn't have that many substitutes. Therefor, an increase in the price of oil should lead to an increase in total spending on oil, meaning that the economy (because it is a net exporter, not a net importer) will, get an increase in net exports.
You could still find, however, that SRAS would potentially decrease, due to the fact that any firms in the country that use (i.e. for transport, productoin etc...) would experience a rise in costs, and so there would be a general rise in the costs of production of most firms in the country, leading to a leftwards shift of SRAS.
Honestly, I am not certain about my logic (hopefully someone comes into correct me if I am talking nonsense lol), but I hope that helps (if its correct).
EDIT: I didn't even answer your question properly hahahaha. No, there wouldn't be a movement along the AD curve.