My arms fell asleep due to the blood circulation being cut off by leaving my arms under my pillow, etc. how long will the pain continue for?

Negative: They're not hurting because the circulation was cut off, it's because the nerve was compressed - it's called "compression neuropathy". The duration of symptoms depends on the degree of damage. The degree of damage is related to how long your arms were compressed. Since we normally move in our sleep, there's generally no serious injury unless you're under the influence of alcohol or drugs.