I'm looking for a precedent of a doctor getting sued after giving his private phone number for a patient to consult, yet not being available at a moment of emergency.
You don't specify what country's law you're interested in, but as you mention precedent, I'll assume you're interested in common-law jurisdictions such as the United States.
The short answer is: you won't find any successful lawsuits such as you describe, at least not without some significant additional facts.
In order to be subject to liability, a person needs to commit a breach of some duty. In a civil suit, this needs to be a duty to the plaintiff.
In order for it to be actionable for the doctor to not answer the phone, the doctor would have to have somehow assumed a duty to answer the phone at that particular time. Giving someone your phone number is not, in and of itself, a promise that you will never take a shower, or go to a movie, or let your battery run down, or for any other reason be unwilling or unable to answer your phone at a moment's notice.
In addition, the patient would have to demonstrate that the doctor, by failing to answer the phone, caused some sort of harm. If the patient is having a medical emergency of some sort, the doctor's advice will almost certainly be: "Hang up and dial 911." This is something the patient can do without the doctor's help.
Without some more significant and compelling facts, no court is going to impose a duty on a doctor to sit by the phone waiting for a patient's call.