As you want to get the details of course and entrance exam which has to be cleared to become Custom Officer so here is the information of the same for you:

I want to tell you that firstly you must have passed the graduation

Secondly you need to give the Civil Services exams

Here for your reference I am giving you the eligibility criteria of Civil Services exams:

Nationality:
A candidate must be either :—
a citizen of India, or
a subject of Nepal, or
a subject of Bhutan, or
a Tibetan refugee who came over to India before 1st January, 1962 with the intention of permanently settling in India, or
a person of Indian origin who has migrated from Pakistan, Burma, Sri Lanka, East African countries of Kenya, Uganda, the United Republic of Tanzania, Zambia, Malawi, Zaire, Ethiopia and Vietnam with the intention of permanently settling in India.

Age Limit:
Candidates must have attained the age of 21 years and must not have exceeded the age of 30 years as on 1st August of the year of examination.

Minimum Educational Qualifications:
Candidates must have passed graduation from any of Universities incorporated by an Act of the Central or State Legislature in India

Number of Attempts:
General: 4
OBS: 7
SC or ST: No Limit

Contact Details:
Union Public Service Commission
UPSC,
Man Singh Road Area,
New Delhi,
Delhi 110069
India

Map Location: