Can I get alcohol poisoning from rubbing hand sanitizer with 70% alcohol on an open small cut?

No. In general the medication in a lotion, gel or liquid you rub on your skin (cut) is not absorbed in enough quantity to be even detected in a blood alcohol test. If you drink a quart of vodka in an hour, the stomach & intestines will absorb it quickly enough for you to get alcohol poisoning. There is a big difference.
NO. No that is not going to happen. Alcohol has to be ingested in a fair quantity to get poisoning.