Why do neurons die so fast without oxygen and nutrients?
Neurons use lot of energy to maintain their polarized state, this is not required to other cells [1,2].

When O2 or blood flow (which is carrying the nutrients) is reduced, the neuronal ATP levels breaks down very fast, with 90% ATP depleted in less than 5 minutes. Without ATP, the neuron can not maintain the correct ion flux, so depolarization occurs causing glutamate excitotoxicity, cell swelling and finally cell death.

1] http://www.acnp.org/g4/gn401000064/ch064.html

2] http://www.scientificamerican.com/article.cfm?id=why-does-the-brain-need-s