Yes, this expression is used, but it is a special-purpose expression.
It is used to emphasis that the thing named is unacceptable. It can be used to sternly forbid someone to do something:
In future you will neither remove nor deface any part of the apartment.
It can also be used to firmly refuse to do something one finds morally repugnant:
I will neither offer up the incense nor say the words acknowledging the divinity of the emperor!
Or to take a firm position in a dispute:
I will neither accede to your demands nor renounce my claims!
It may also have mundane uses, though it is considered more formal and comes across as emphatic:
Sorry, we have neither cream nor sugar for your coffee.
Ordinarily we would use "don't" as a negative auxiliary verb and phrase that as:
Sorry, we don't have any cream or sugar for your coffee.
The neither, nor construct can also be used to emphasis impossibility. For example, the unofficial motto of the US Postal Service:
Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds.
Your example is weird because it implies one of two things:
Your client has told you that it is extremely important that you not contact them before midday. You are using this special construction to show that you fully understand that contacting them before midday would be absolutely unacceptable to them and that you will be very careful not to do so.
Your client has repeatedly demanded that you answer before midday, but you refuse, and now you are using this construction in a last attempt to convey to them that you consider these demand unreasonable and offensive and will not comply.