What should a customer service executive do if a customer is very annoyed and the customer service manager can't be reached?

DON'T apologies!! customer will use that against you the company you representing! respectfully say: Sir/Ma'am i understand the case, and i am doing everything i can do regarding your case, and i will sure refer it the relevant department so they can take care of it.
Please be patient and everything should be resolved! and don't say the manager is out!!! they will use it against as well.
how is that?

In this case he may try to solve the problem with his given limits, If the issue need more permissions, He must make it clear to the customer that he must wait the manager reply.
It will depend on the given limits the customer executive must not exceed.

When the manager is out of contact, the first approach for the agent is to look at a senior best agent who is available on the floor and get him/her to the customer and resolve the issue with lot of courtliness and assurance.
If the agent does not find any senior person, then try to bring some colleague who belongs to the same department and provide an assurance on resolving the issue (or) making the customer understand about the issue.
The reason I ask the agent to look for someone else is because the customer is already irritated with the him/her and the customer will never agree to his/her apology (or) the resolution that is provided, when the agent brings in another person, the effect on that person is much more milder and the customer is ready to listen to him.
(but in this there is a catch - The agent should give a complete background of the issue to the other agent, so that he/she does not ask the customer on the issue (or) provide an incorrect information and the situation can worsen up)

First thing and most important is to listen to the client attentively and make him fee that his case is the center of your attention and you fully understand...if the CS Manager is not available and there is no decision maker that will replace him then give him time limit that you will be able to respond to him.
However I preferably suggest to find a branch Manager or other leading person to just talk to the client even if it wont solve the problem ..
it will make him feel valued and might calm him a little.

In the case the CS Manager is absent, don't make yourself a hero.
Refer the cutomer to a branch manager or someone senior who is more experienced in these cases.
Anyway, it should be smooth and comforting process for the customer, and remember each good voice will bring 10 times more satisfied customers.

he/she should assure the customer that there will not be any problem from that time and he/she should solve that problem in the absence of senior on the basic of his/her filed of authorization.

When the CS Manager is not around, the CE should be confident, listen to the customer and jot down complaints, make him feel as a king and let him know he is important. Get his details and assured him someone from the top management will get back to him ASAP. Follow up to make sure someone get back to the guest,

assure the customer that problem will be solved and take full details and brief from the customer including latest contact details and pass to the managers with URGENT in the heading.
always make the customer case important as that is the reality...
customers looking for someone to listen and help instead of arguing and telling them that they have no right to chase and complain...
"Customers always right".

Apologies , And Listen Carefully For The Customer Complaint And Write it Infront Of the Customer , Then Ask Him For His Phone No.
And His Name , And Telling Him Very Clear That You Will Call Him In The Morning Or In The Evening , And Give Him an Exact Hour ,So He Feel A Little Relief And Don't You Ever Dare To Forget Him !!!

- Don't interrupt.
- Don't take the criticism (judgment) personally.
- keep irate customers out of the public eye.
- Don't blame.
- Don't argue.
- Be warm and friendly.
- Be professional.
- No false promises.

you will say i tottaly understand your situtation Mr.X , and i will rely the message to my superiors as they will be contacting you very soon , and write down his contact details .