How do neurons receive the ions needed for creating electrical pulses?
Short answer
The sodium-potassium pump in the epithelial cells lining the brain capillaries pumps Na+ and K+ into the brain.

Background
The blood-brain-barrier is formed by the endothelial cells lining the capillaries in the brain. Not many compounds can pass this lining, especially not when they are hydrophilic, such as charged metal ions. However, as you rightfully state, somehow these metal ions, and especially Na+ and K+ need to get access to the brain because they are crucial for membrane potentials and action potentials, and hence for neuronal survival and function. And indeed they can, albeit the exchange between blood plasma and brain is slow compared with other tissues. 

Intravenously administered K+, for example, exchanges with muscle K+ in 1 hr, but K+ exchange in brain is only half completed in 24 to 36 hr. Na+ exchange is somewhat faster, with half-exchange into brain occurring in 3 to 8 hr. Despite its relatively slow entry into the brain, Na+ exchange across the blood—brain barrier appears to occur by mediated transport. This occurs, in part, through brain capillary Na+,K+-ATPase (the sodium-potassium pump). The Na+,K+-ATPase in the brain also may mediate removal of interstitial fluid K+ from the brain and thereby aid in brain K+ homeostasis. 

Reference
- Laterra et al. In: Siegel et al. eds. Basic Neurochemistry: Molecular, Cellular and Medical Aspects. 6th ed. Philadelphia: Lippincott-Raven; 1999