To become custom officer you have to pass Civil service examination organized by UPSC.. here I am giving you information about required eligibility for civil service examination below :

The Civil Services Examination comprises:
(i) Civil Service Aptitude Test (CSAT)
(ii) Civil Services Main Examination
Personality Test

Eligibility Criteria:
You have to passed graduation in any discipline from a well recognized university.
Age limit : The candidate must be between 21-30 years of age.

Important date :
Civil Services Preliminary Exam 2014
Notification Date : 17th May 2014
Last dates for Applications : 16th June 2014
Examination will be held on : 24th August 2014

Scheme of CS (Preliminary) Examination
The Preliminary Examination consists of two papers of objective type maximum of 400 marks.
General Studies Paper - I 200 Marks
General Studies Paper - II 200 Marks

Syllabus of Paper I -(200 marks) Duration : Two hours

Current events of national and international importance.
History of India and Indian National Movement.
Indian and World Geography-Physical, Social, Economic.
Geography of India and the World.
Indian Polity and Governance-Constitution, Political.
System, Panchayati Raj, Public Policy, Rights Issues, etc.
Economic and Social Development-Sustainable.
Development, Poverty, Inclusion, Demographics, Social.
Sector Initiatives, etc.
General issues on Environmental ecology, Bio-diversity and Climate Change - that do not require subject specialization and Climate Change - that do not require subject specialization.

Syllabus for Paper II-(200 marks) Duration: Two hours
Interpersonal skills including communication skills;
Logical reasoning and analytical ability
Decision making and problem solving
General mental ability
Basic numeracy (numbers and their relations, orders of magnitude, etc.) (Class X level), Data interpretation (charts, graphs, tables, data sufficiency etc. - Class X level)